


H 






LIBRARY OF CONGRESS. 



Chap, Copyright No,. 

Shelf„JF_0 



UNITED STATES OF AMERICA. 



rs 



A BRIEF INTRODUCTION 



TO 



THE INFINITESIMAL CALCULUS 



•T^Q^i 



A BRIEF INTRODUCTION 



Infinitesimal Calculus 



DESIGNED ESPECIALLY TO AID IN READING 

MATHEMATICAL ECONOMICS AND 

STATISTICS 



BY 



IRVING FISHER, Ph.D. 

Assistant Professor of Political Science in Yale University 
Co-author of Phillips's and Fisher's " Elements of Geometry ' 



Ntfo gorfc 
THE MACMILLAN COMPANY 

LONDON: MACMILLAN & CO., LTD. 

1897 

All rights reserved 



v 




Copyright, 1897, 
By THE MACMILLAN COMPANY. 



J. S. Cushing & Co. — Berwick & Smith 
Norwood Mass. U.S.A. 



PREFACE 

This little volume contains the substance of lectures by 
which I have been accustomed to introduce the more 
advanced of my students to a course in modern economic 
theory. I could find no text-book sufficiently brief for my 
purpose, nor one which distributed the emphasis in the 
desired manner. My object, however, in preparing my 
notes for publication has not been principally to provide a 
book for classroom use. It must be admitted that very few 
teachers of Economics as yet desire to address their stu- 
dents in the mathematical tongue. I have had in mind not 
so much the classroom as the study. Teachers and students 
alike, however little they care about the mathematical 
medium for their own ideas, are growing to feel the need of 
it in order to understand the ideas of others. I have fre- 
quently received inquiries, as doubtless have other teachers, 
for some book which would enable a person without special 
mathematical training or aptitude to understand the works 
of Jevons, Walras, Marshall, or Pare to, or the mathematical 
articles constantly appearing in the Economic Journal, the 
Journal of the Royal Statistical Society, the Giornale degli 
Economisli, and elsewhere. It is such a book that I have 
tried to write. 

V 



vi PREFACE 

The immediate occasion for its publication is the appear- 
ance in English of Cournot's Principes mathematiques de la 
theorie des richesses, in Professor Ashley's series of " Eco- 
nomic Classics." The " non-mathematical " reader can 
only expect to understand the general trend of reasoning in 
this masterly little memoir. If he finds it as stimulating as 
most readers have, he will want to comprehend its notation 
and processes in detail. 

I have tried in some measure to meet the varying needs 
of different readers by using two sorts of type. If desired, 
most of the fine print may be omitted on first reading, and 
all on second. The reader is, however, advised not to pass 
over all of the examples. 

Although intended primarily for economic students, the 
book is equally adapted to the use of those who wish a short 
course in "The Calculus " as a matter of general education. 
I therefore venture the hope that teachers of mathematics 
may find it useful as a text-book in courses planned espe- 
cially for the " general student." I have long been of the 
opinion that the fundamental conceptions and processes of 
the Infinitesimal Calculus are of greater educational value 
than those of Analytical Geometry or Trigonometry, which 
at present find a conspicuous place in our school and college 
curricula. Moreover, they are almost as easily learned, and 
far less easily forgotten. 

IRVING FISHER. 

New Haven, September, 1897. 



CONTENTS 



CHAPTER I 

PAGE 

The General Method of Differentiation i 

CHAPTER II 
General Theorems of Differentiation . . . .16 

CHAPTER III 
Differentiation of the Elementary Functions . . .30 

CHAPTER IV 

Successive Differentiation — Maxima and Minima . . 37 

CHAPTER V 
Taylor's Theorem 49 

CHAPTER VI 
Integral Calculus 57 

APPENDIX 
Functions of More than One Variable ... 73 



INFINITESIMAL CALCULUS 



CHAPTER I 

THE GENERAL METHOD OF DIFFERENTIATION 

1. The Infinitesimal Calculus treats of the ultimate ratios 

of vanishing quantities. This definition, however, can only 
become intelligible after some actual acquaintance with 
" ultimate ratios." 

2. The conception of a limiting or ultimate ratio is funda- 
mental in many familiar relations. It is impossible, without 
it, to obtain a clear notion of what is the velocity o£ a body 
at an instant. The average velocity of the body during a 
period of time may readily be defined as the quotient of the 
space traversed during that period divided by the time of 
traversing it. If a steamer crosses the Atlantic (3000 miles) 
in 6 days, we may say that the average speed is 3000 -+■ 6, 
or 500, miles per day. But this does not tell us the speed at 
various points in the voyage, under head winds, storms, or 
other conditions, favorable or unfavorable. What, for 
instance, was the speed at noon of the third day out? We 
may obtain a first approximation to the desired result by 
taking the average speed for a short time after the given 
instant ; that is, taking the ratio of the distance traversed 



2 INFINITESIMAL CALCULUS 

during (say) the following hour to the time of traversing it, 
which is -oV of a day. If this distance be 20 miles, we obtain 
20^21, or 480 miles per day, as the average speed during 
that hour. For a second approximation we take a minute 
instead of an hour ; for a third, a second instead of a minute, 
and so on. The ratio of the space traversed to the time of 
traversing it becomes closer and closer to the true speed. 
Though both the time and space approach zero as limit, 
their ratio does not. The limit which this ratio approaches, 
or the ultimate ratio of the distance traversed to the time of 
traversing it when both distance and time vanish, is the pre- 
cise speed at the instant. 

3. Let us apply this method of obtaining velocity to 
bodies falling in a vacuum. We know from experience that 
the distance fallen equals sixteen times the square of the 
time of falling, i.e. s = 16 t 2 , where s is the distance fallen 
from rest (measured in feet), and / is the time of falling (in 
seconds). Consider the body at some particular instant, / 
being the time to this particular point and s the distance. 
Suppose we wait until the time has increased by a small 
increment At, during which the body increases its distance 
from the starting-point, s, by the small increment As. Since 
the above formula holds true of all points, it holds true now, 
when the time is /+A/, and the distance is s+As. That is, 

s + As = 16 (/ + At) 2 . 

This gives 

s + As=i6t 2 +32l- A/+ i6(A/) 2 . 

But s =i6t 2 . 

Subtracting, we have 

Ai-=32/. At+ i6(A/) 2 , 



GENERAL METHOD OF DIFFERENTIATION 3 

As 

whence — =32^+16 A/. (1) 

This is the average velocity during the small interval At. 

Thus, if A^ = \ second and t be 5 seconds, the average speed of the 
body during that half second (viz., the one beginning 5 seconds from 
rest) is 32 x 5 + 16 x 4-, or 168 feet per second. If we take t ^q of a sec- 
ond instead of J, we have 32 x 5 + 16 X t Jq-, or 160.1 feet per second. 

The speed at the very instant of completing the 5 th 
second is obtained by putting A/=o, which gives 160 as 
the instantaneous speed. 

Now, when A/— o, we call it dt, because o would not 
remind us of the kind of quantity which vanished, whereas 
dt does suggest /, the magnitude which vanished. When At 
becomes zero, or dt, As evidently becomes zero too, for a 
body cannot go any distance in no time. This zero we call 
ds^~ Equation (1) therefore becomes at the limit 

— — 3 2 t + dt, 
dt * . ' 

or - = 32/4-0, 

o 

ds 
which may be written — = 32 /, (2) 

for we can neglect the zero on the right, but not those on 
the left (the ratio of two zeros does not reduce to zero). 

Thus, the velocity of a body which has fallen 4 seconds is 32 x 4 
or 128 feet per second. What velocity has it after falling 10 seconds ? 
After 7 seconds ? 3 ? I J ? Zero ? 



4. It may be objected to the reasoning in the last article 

that — or - is indeterminate. This is true. - is equal to 
dt o o 

2, or 19, or 1, or any number we please. But the limit of 



4 INFINITESIMAL CALCULUS 

As . . ds 

— is not indeterminate. We thus use — in two distinct 

At dt 

senses, viz. : ,. As , lim As 

' lim — and 

At lim At 

The first is determinate, the second is indeterminate, 
though for that very reason it may always be put equal to 

As 

the first. Only the first, or lim — , is important. This is 

the "ultimate ratio" of two vanishing quantities. 

5. The zeros ds, dt, and others obtained by the vanishing 
of finite quantities are called infinitesimals or differentials. 
They are, however, true zeros and not simply very small 
quantities. Before the distinction between the limit of the 

ratio of vanishing quantities ( e.g. lim — ), and the ratio of 

f lim AA ^ A/ ' 

their limits [ e.g. , was completely grasped, the meth- 
yl lim At) 

ods of the Calculus were viewed with suspicion. Bishop 
Berkeley derisively called differentials the "ghosts of de- 
parted quantities " about which only shadowy conclusions 
were possible. 

The ultimate ratio of the vanishing quantities As and At, 

i.e. the magnitude lim — or — , is called the differential 
At dt 

quotient or differential coefficient or, better, derivative of s 
with respect to /. Since s is the same as 16 t 2 , we also say 

ds • 

that — , or its equal 32 /, is the derivative of 16 /-. 
dt 

6. In the particular case considered above, the differ- 
ential quotient is a velocity and may be denoted by v. 
Equation (2) thus becomes * v = 32 t. 

* If distance be measured in. centimetres instead of in feet, we 
should have v = 980/, and in general v = gt, where g is a constant 



GENERAL METHOD OF DIFFERENTIATION 5 

Velocity at a point may now be denned as the ultimate 
ratio of the space traversed just after passing the point to the 
time of traversing it when the space and time approach zero 
as limit 

7. Examples. 

1. What is the velocity of a body which has fallen 10 seconds? 
100 seconds ? 1 J seconds ? 

2. What is the velocity of a body which has fallen 16 feet ? 
Hint. — First find how many seconds it has fallen by using s=i6f 1 . 

3. What is the velocity of*a body which has fallen 64 feet ? 4 feet ? 
1 foot ? 2 feet ? 

4. It being known that a body, falling not from rest, but with an 
initial velocity of 5 feet per second, obeys the law 

s = 16*2 + 5/, ( r ) 

what will be its velocity at the end of any time t ? 

Hint. — Let t receive an increment At, causing s to increase by As, 

so that 

s + As = l6(t + At) 2 + 5(* + At). (2) 

Subtract (1) from (2), divide by At and then reduce At and As to dt 

and ds. A ds _ , . „ 

Am. — = \2t + K. 
dt * * 

5. W T hat will be the velocity at the end of 10 seconds ? At the end 
of 69 feet ? 

6. It being known that a body falling with an initial velocity of u 
obeys the law s = ^gt 2 + nt, what will be its velocity at the end of 
time t ? When / = 3 ? 

8. When one quantity depends upon another, the first is 
said to be a function of the second. A change in the second 
is accompanied by a change in the first. 

depending for its numerical value on the units chosen for measuring 
space and time. 



6 INFINITESIMAL CALCULUS 

Thus the distance a body falls from rest is a function of the time of 
falling, for how far the body falls depends on how long it has fallen; 
the demand for an article is a function of its price, for if the price 
changes the demand changes; if y = x 2 , then y is a function of jr, for 
a variation in the magnitude of x necessitates also a variation in the 
magnitude of y. 

9. When one quantity is a function of another, the latter 
is called the independent variable, and the former the de- 
pendent variable. 

The distinction between the independent and the depend- 
ent variable is only for convenience of expression. The 
two may be interchanged. 

Thus, as the distance of a falling body from the starting-point 
changes, there is also a change in the time it has taken. Hence we 
may say that " time of falling " is a function of " distance fallen." Simi- 
larly price may be regarded as a function of demand. Again, y = x 2 
may be written x = Vy, thus making x a function of y. The idea of 
functional dependence is therefore quite different from that of causal 
dependence. Functional dependence is a mutual relation. 

In the example of falling bodies s was a function of /, and 
what we accomplished was to find the differential quotient 
or derivative of that function. The derivative in this case 
was a velocity. In general the process of finding the differ- 
ential quotient of any given function is called differentiation, 
and is the subject matter of the Differential Calculus, one 
of the two branches into which the Infinitesimal Calculus is 
divided. The Differential Calculus will occupy us in the 
first five chapters of this book. 

10. A second important application of the idea of a differ- 
ential quotient of a function is to the tangential direction of a 
curve at any point on it. The Calculus enables us to conceive 
in the most general manner of a tangent to a curve. The 



GENERAL METHOD OF DIFFERENTIATION 



student should observe that the usual definition of a tangent 
to a circle will not apply to any and all curves. A straight 
line may have only one point in common with a curve and 
yet cut it and not be tangent. 

II. Let PS be a curve whose equation is 

y=i + 5* — x 2 . (i) 

That is, for any point P upon it, the " ordinate," y (or dis- 
tance, PA, from that point to the horizontal axis), is related 




Fig. i. 

to the "abscissa," x (or distance, OA, from the vertical axis), 
in the manner expressed by (i). PA is a function of OA ; 
i.e. the height, PA, of any point P on the curve depends 
upon its distance, OA, from the vertical axis. 

What is the direction of the curve at the point P ? The 
direction from the point P to another point P' is the direc- 
tion of the secant line QPP'. The point P' has for abscissa, 



8 INFINITESIMAL CALCULUS 

x + Ax, and for ordinate, y + Ay. Since the relation (i) 
holds true of all points on the curve, it holds true of P\ 

Hence y + Ay = i + 5 (x + Ax) — (x -f- Ax) 2 , 

or y + Ay=i + 5x+$Ax — x 2 — 2 ^ A^ — (A*) 2 . 

Subtracting y = 1 + 5 x — x 2 , 

we have Ajy = 5 A^ — 2 .# Ax — (A^) 2 , 

whence -^- = z — 2 x — A^*. 

A* ° 

We may pause here a moment to see what this result 

means. ^ or ?-£- is the " slope " of the line QPP\ That 
Ax PC l ^ 

is, it is the rate at which a point moving from Q toward P ! 
rises in proportion to its horizontal progress. It is the same 
sort of magnitude as that referred to as the " grade" of an 
uphill road which rises " so many feet to the mile (hori- 
zontally)." If -£- = — , QPP } fises one foot in every ten 
JJ Ax 10 J 

horizontally. The " slope " of a line shows its direction. 

Ay 
The equation — = 5 — 2 x — Ax shows that the " slope " of the 

secant line Q'PP' is to be found by taking 5 and subtracting, first, 
two times the number of units in OA and then the number of units 
in AB. For instance, if OA = 2 and AB — J, then 

Ax D 2 2 > 

i.e. the secant slopes 1 fc >t up for every 2 feet sidewise. 

12. But we have not yet reached the tangent at P. Let 
the point P be gradually shifted along the curve toward P 
until it ultimately coincides. The secant QP ] will gradually 



GENERAL METHOD OF DIFFERENTIATION 9 

change its direction and approach a limiting position QP. 
This liiniting position we call the tangent. Its slope is 

dy 

Thus, if #(i.e. OA) is 2, ^ = 1. That is, QP is inclined at 45 . 
dx 



If x is 4, 



dx 



3 ; i.e. the curve slopes down, not up. 




Fig. 2. 
Examples. 



-A, positive slope; B, zero slope; C, negative slope. 



1. What is the slope of the tangent to the above curve at the point 
whose abscissa is 1 ? o ? 2 J ? What does the answer to the last 
mean ? 3 ? What does this mean ? 6 ? — 1 ? 

2. Derive the formula for the slope of the tangent to the curve 
y = 1 -f x + x 1 . 

13. To construct a tangent at P, all we need to do is to 
draw a line through .Pwith the required slope. Thus, if we 
wish the tangent to the point whose abscissa is 1, we find 
from the above formula that its slope is 3. We therefore 
lay off a horizontal line LM (Fig. 1) equal to one unit, and 
at its extremity erect a vertical, MN } equal to three units. 
Draw LN' y this has the required direction. Then through 
P draw a line parallel to LN. This will be the tangent. 

The problem of drawing a tangent and calculating its slope was 
one of the chief problems which gave rise to the discovery of the 
Calculus. 

14. It is evident that we could approach P from the left as well 
as from the right. We should^ however, reach the same limiting posi- 



10 INFINITESIMAL CALCULUS 

tion unless there should be an angle in the curve at the point P as in 
Fig. 3. In this case, the progressive (PK} and regressive {HP) tan- 
gents do not coincide. 




Such peculiar points are not considered in this little treatise. All 
the functions are such that, for the values of the independent variable 
which are considered, the progressive and regressive derivatives are 
identical. The curves considered are all "smooth," that is, have no 
angles or sudden changes in direction. In many applications of the 
Calculus, such as to statistical or economic diagrams, it is often con- 
venient first to smooth out the curves considered. When we want to 
see from a plot of the population what is the general rate of increase, 
we draw a tangent not to the plot of the actual figures, but to a smooth 
curve coinciding as nearly as possible with the plot. 

The student will be able to satisfy himself in every particular case 
to be considered that the progressive and regressive derivatives are 
identical. 

Thus, for s=i6t 2 in section 3, let t receive a decrement A ! t, causing 
s to have a decrement A^. Then 

s - A's= i6(7- A'/) 2 . 

Expanding, subtracting, and dividing as before, we obtain 

— = 32/-i6aV, 

A f t 

which reduces at the limit to 

d's 

— — 22 1, as before. 
d't * 

Indeed, we assume in general, that it is physically impossible for 
a. body to change its velocity per saltum. Hence the definition of 



GENERAL METHOD OF DIFFERENTIATION 11 

velocity given in section 6 is equivalent to the following alternative 
definition : the ultimate ratio of the space traversed just before reaching 
the point to the time of traversing it when the space and time ap- 
proach zero as limit. 

We shall, therefore, henceforth treat only of functions whose deriva- 
tives are continuous and which are themselves continuous, that is, 
which in changing from one value to another, pass through all inter- 
mediate values. 

15. We have seen that the conception of an ultimate ratio 
clears up the notion of velocity in mechanics and tangential 
slope in geometry. It is also applicable to much else in 
both these sciences as well as in all mathematical sciences. 
Momentum, acceleration, force, horsepower, density, curva-- 
ture, marginal utility, marginal cost, elasticity of demand, 
birth rate, " force of mortality," are all examples. 

The conception of an ultimate ratio or of the derivative of 
a function is not dependent; however, on any special applica- 
tion. It is purely an abstract idea of number. 

16. Thus let two variables x and y fulfil the equation 

y = x n , 

where n is a constant and a positive integer. We may 

obtain the differential quotient -+- for any particular value 
of x, as follows : 

Let x receive an increment Ax producing an increment of 
y denoted by Ay. Then, by the binomial theorem, 

y + Ay = (x + Ax) n , 

= x n + nx^Ax + n ( n ~ T ) x n~2 ^ x y 
2 
+ ••• + (A*)" 

= x n + nx n ~ 1 Ax + Ax? (•••). 



12 INFINITESIMAL CALCULUS 

Subtracting y = x n , 

we have Ay = nx n ~ 1 Ax + (A.#) 2 (---). 

Whence -^- = nx* 1 ' 1 + Ax ( ~ •), 

Ax K ' 

where the parenthesis is evidently a finite quantity and re- 
mains finite after Ax becomes zero. Hence, when Ax 
becomes zero, the term Ajt(--«) becomes zero, and the 
equation becomes, 

dx 
17. This is the first and most important specific formula 
which we have reached for the derivative of a function. It 
states that, to obtain the derivative of x n , a power of x, we 
need only reduce the exponent by unity and use the old 
exponent for coefficient. 

Thus the derivative of x z is 3 x 2 . When x passes through the value 
2, $x 2 becomes 12; that is, y, or x s , is increasing 12 times as fast as x. 

dv 

-j- is the rate at which y increases compared with the rate we make x 

increase. If y denotes the distance of a moving body from the start- 
ing-point, and x denotes the time it has moved, — , or 3 x 2 , expresses 

its velocity. Again, if x and y are the "coordinates" (i.e. the " ab- 
scissa" and "ordinate") of a curve whose equation is y = x s , then 
3 x 2 is its slope at the point whose abscissa is x. 

Although it is logically unnecessary, it is practically helpful to pict- 
ure the differential quotient as a possible velocity or a possible slope. 
Of the two independent discoverers of the Calculus, Newton seemed 
to have employed the former image, and Leibnitz the latter. New- 
ton's term for a differential quotient was " fluxion." 

Examples. — 1. Find the derivatives of x 12 , x 5 , x 2 > x. What is the 
meaning of the answer to the last ? 

2. How many times as fast does y increase as x when y = x^ and 
x is 2 ? 

3. How fast does x Q increased compared with x when x is — 1 ? 
What does the negative answer mean ? 



GENERAL METHOD OF DIFFERENTIATION 13 

1 8. The process employed in this chapter for obtaining 
the derivative of a function is called the " general method 
of differentiation." It consists (i) in giving to the inde- 
pendent variable a small increment, thus causing another 
small increment* in the dependent variable or function; 
(2) in writing the relation between the two variables first 
without and then with these increments and subtracting the 
first from the second; (3) in dividing through by the incre- 
ment of the independent variable ; (4) in reducing both 
increments to differentials in the last equation. 

This process should be thoroughly mastered by the 
student, for it contains, in embryo, the whole of the Infini- 
tesimal Calculus. 

He will observe that the order of steps (3) and (4) cannot 
be inverted without producing the barren result = 0. 

19. Nevertheless, we can anticipate the result of step (4) 
without changing from the form of (2). Thus, the equation 

yields at step (2) : 

Ay=2 Ax-\-6x Ajt: -f- 3 (Ajc) 2 +15 x 2 Ax+i$ x(Ax) 2 +$(Ax) s 

= (2 + 6*+ i5* 2 )A*+(3+ i5*)(A*) 2 + 5(A*) 3 . 

It can readily be foreseen that step (3) (i.e. dividing by 
Ax) will remove the first Ax, and reduce the exponents of 
the powers of Ax by one, and that therefore when step (4) 
is performed (i.e. reducing Ax to zero), all terms beyond 
the first will disappear, leaving 2 + 6^+15 x 2 as the 
derivative. Now it is clear that this result could have been 
anticipated simply by neglecting the terms i?tvolving powers 

* Decrements may always be regarded as negative increments. 



14 INFINITESIMAL CALCULUS 

of Ax higher than the first, and taking the coefficient of the 
first power as the required derivative. 

Though this process of neglecting certain terms at step 
(2) is a mere anticipation of what must necessarily happen 
at step (4), it may be shown to be perfectly natural in situ. 
If Ax be less than one, (A.*) 2 will be less than Ax, and 
(Ax) s less than (Ax) 2 , etc. By making Ax smaller and 
smaller, the higher powers (Ax) 2 , (Ax) s , etc., can be made 
indefinitely small, not only absolutely, but in comparison 
with Ax. At the limit, Ax becomes the infinitesimal dx, 
(Ax) 2 becomes (dx) 2 , an infinitesimal of the second order; 
(Ax) s becomes (dx) s , one of the third order, etc. All infini- 
tesimals are true zeros, but those of the second order are 
infinitely small compared with those of the first order, and 
those of the third infinitely small compared with those of 
the second, etc. By such comparisons we simply mean 

that ^ — L i n the sense of lim ^ — '— is zero (which is evi- 

dx Ax 

dently correct, since the latter expression is identical with 

lim Ax) ; that x ' in the sense of lim^ — '— is zero, etc. 

J} (dx) 2 (Ax) 2 

The higher powers of Ax thus growing negligible relatively 
to Ax, the terms in which those powers occur as factors 
must also grow negligible (provided, of course, the other 
factor composing each such term does not approach infinity 
as limit). 

Thus, if Ax is t Jq, (A*) 2 is T oJo o> and ( A *) 3 onl y nnrfonnr Con- 
sequently in the equation 

Ay = (2 + 6*+ i5^ 2 )A^+(3 4- i5*)(A.r) 2 + 5( A ^) 3 > 

we can, by reducing Ax sufficiently, make the terms beyond the first 
as small as we please compared with the first, no matter what be the 
value of x, so long as it is finite thus keeping the parentheses finite. For 
instance, if x be 2, we have Ay = 74 Ax + 33(A.r) 2 + 5(A#) 3 . 



GENERAL METHOD OF DIFFERENTIATION 15 

Then, if 

Ax be .01, this becomes 

Ay - .74 + .0033 + .000,005. 
If Ax = .001, it becomes 

Ay = .074 + .000,033 + .000,000,005. 
If Ax = .000,001, it becomes 

Ay = .000,074 + .000,000,000,033 + .000,000,000,000,000,005, 

and the smaller we make Ax, the more negligible become the terms 
involving (Ax)' 2 and (A.*-) 3 , until at the limit they become, not simply 
negligible " for practical purposes," but absolutely negligible. 

The anticipatory neglect of terms involving powers of Ax 
higher than the first often saves a great deal of labor. 

Examples. 

1. Find -2- when y = x b . 

dx 

2. Find -£• when y = x~ + 8 x Q -f 4. 

dx 

3. Find^ when;'= 10 * 100 . 

dx J 

4. Find -2- when y = ax m -f bx n . 

dx 



16 INFINITESIMAL CALCUL US 



CHAPTER II 

GENERAL THEOREMS OF DIFFERENTIATION 

20. If we differentiate 

y = 2x 
by the general method, we obtain 

t- « 

Clearing this equation of fractions, we have 

dy = 2 dx. (2) 

This latter equation is to be taken in the same sense as the 
former. Both state the fact that y increases twice as fast 
as x. 

Now the student will object at once that, since dy and dx 
are zero, the last equation, though true, is no truer than 

dy = 3 dx. 

This is correct enough, and yet we cannot employ the 
latter equation to show the rate at which y increases com- 
pared with x when y and x are connected by the relation 

y= 2 x. 

Why not ? To answer this question we must recur to the 

double meaning of -^- (shown in § 4). 
dx 



GENERAL THEOREMS OF DIFFERENTIATION 17 

Ay ' 

The meaning lim-^is the determinate and important one 

Ax 

, . lim Ay 
and not *-• 

lim Ax 

Although the equation dy = $dx is true, and it may be 
written ij m ± y = 3 n m & Xf 



or also 



lim A)' 
lim Ax 



yet the left member is indeterminate, and therefore not cer- 
tain to be the same as 

litn^L 

Ax 

Of course the valid equation 

dy = 2 dx 
itself yields in like manner 

lim Ay __ 
lim A^ 

with an indeterminate left member. But then we know 
beforehand, from actual differentiation, that 

lim — = 2. 
Ax 

Equation (2), therefore, though it does not properly lead 
back to equation (1) from which it was derived, is to be con- 
sidered simply as an elliptical form of equation (1). 

Thus, dy = 6xdx is to be taken in the sense 



dy _ 
dx 

Ax 
6 x is a differential quotient and 6 xdx is a differential. 



- = 6x, 
dx 



which in turn means lim — ^- = 6 x, 

Ax 



18 INFINITESIMAL CALCULUS 

These conceptions are strictly correlative. To obtain the differen- 
tial quotient from the differential, we simply drop the dx ; to obtain 
the reverse, we add it on. 

Examples. 

1. What is the differential of x 5 ? 

2. The differential quotients of x 7 , x 10 , x* ? 

21. To express the mere fact that y is a function of x, 
without specifying exactly what function, it is customary to 
use the letters F, f, <£, \p (and rarely others) followed by x 
in a parenthesis. They maybe regarded simply as abbrevia- 
tions of the word " function." Thus 

y = Function of x 
is abbreviated to y = F(x). 

It is to be observed that the letters F, f, <£, \f/ 9 etc., do not repre- 
sent quantities like x and y f but, like A and d, represent operations 
on quantities. 

22. The general expression for a function, such as <£ (x), 
is often used to express, within brief compass, any special 
function. Thus if we have the equation 

i +x — .6x f +-^- 
ax n 



S x 



i + x 2 4X 2 

We may shorten this to y = <f> (x) by denoting the clumsy 
right-hand member by <£ (x). 

Again, if we have a definite curve, such as a statistical 
diagram, whose coordinates we call x and y, we may use 

y=f(x) 
to express the fact that y is related to x in the particular 
manner delineated by the curve. 



GENERAL THEOREMS OF DIFFERENTIATION 19 

23. The differential quotient, or derivative of a function 
of x, is itself a function of x. 

To denote the differential quotient of 

Fix), 

we use the expression F ! (x). 

Thus let (p(x) stand for x 6 . 
Then (p'^x) stands for 6x b . 

The differential oi F (x) is therefore expressed by 

F\x)dx. 

24. Another method of expressing the differential quo- 
tient of 

Fix) 

connects it with the general method of differentiation. Thus, 
if x receives an increment Ax, F(x) will become 

F(x + Ax). 

This differs from its original value F(x) by 

F(x + Ax) — F(x). 

The ratio of this increment of the function to the incre- 
ment Ax, of the independent variable x, is 

F{x + Ax) — F(x) 

Ax 

Its limit, viz. lim — ^ 1 ±-2 9 

Ax 

is the differential quotient of F(x) ; i.e. is 

F'(x). 

The above process is identical with the general method of differen- 
tiation, though we have expressed it without the use of y. We might 
have proceeded as follows : 



20 INFINITESIMAL CALCULUS 

Put F(x) equal to y so that 

y = F{x). 

Subtract this from y + Ay = F(x -f A.*), 

and divide by A;t, giving 

Ay _ ^Q + A*)- ^(» 
A^~ A;c 

or, at the limit 

dy_ = Hm ^(^ + A^)-^(^) 

25. Yet one more notation should be familiarized. 
Rather it is a new application of an old one. Instead 

of writing -^-, we may replace y in this expression by 
dx 

Fix), so that it reads 

dx 

The student will do well now to release his mind from y as any 
necessary element in the analysis. It is to be regarded merely as a 
further abbreviation of F(x). 

F(x) rather than y is to be thought of as primarily the function of 
(V). Thus, in our introductory example, instead of denoting space by 
s and writing s = 16 / 2 , we need only say if t denotes time, the function 
of /, 16/ 2 , will denote space. 

So also if x denotes the abscissa of a curve, F(x) instead of y de- 
notes its ordinate. 

Thus, v ; is 2x, 

dx 

or d (x 2 ) = 2 xdx. 

Examples. — \ J = ? d (x±) = ? 

dx 

We thus have four methods of denoting the differential 
quotient of y, or its equal F(x) ; viz. : 

dy d[F(x)\ ,, F(x + ±x)-F{x) 

'dx'' ~Tx ' ^ W ' u A* 



GENERAL THEOREMS OF DIFFERENTIATION 21 

26. If a function of x is the sum of several functions of 
x. i.e. if 

then, since this equation holds true of all values of x 9 it 
holds true when x becomes x + Ax. so that 

Fix + Ax) = ft (x + A^) +f 2 (x + Ajc) + • • .. 

Subtracting the upper equation from the lower, and divid- 
ing by Ax, we obtain 

F(x + Ax) — F(x) ___fi(x + &x) —fi(x) 

Ax Ax 

f 2 (x + Ax)-f 2 (x) 
" r "" A* +'"' 

or at the limits F \x) = ft 1 \x) + f 2 \x) -\ . 

That is, the differential quotient of the sum of several func- 
tions is the sum of the differential quotients of those functions. 
The same reasoning establishes the corresponding theorem 
for the differe?ice of functions. 

Thus the differential quotient of x 2 -f x z is 2x + 3x 2 . 
Sometimes the theorem is used in its elliptical form 

F' O) dx =fi'(x) dx +fj(x) dx + .., 
or again F' (x) dx = \_f\ r (x) + f 2 ' (x) + •••] dx. 

Examples. — Find the differential quotient of: 
1. x* + x 2 - x±. 2. x 1 - x 2 + x. 3. - x 2 -f x 10 . 

27. If a function of x is the sum of another function of 
x and a constant quantity, i.e. if 

F(x)=f(x) + K, (1) 

where K is a constant, then 

F(x)=f( X ), (2) 



22 INFINITESIMAL CALCULUS 

the same result as if K were not present in (i) at all. The 
proof of (2) is simple. When x becomes x + Ax, (1) be- 
comes 

F(x + Ax)-/(x + kx)+K, (1)' 

When we subtract (1) from (1)', K disappears entirely, and 
we have, after dividing by Ax, 

Fix + Ax) -F(x) _ fix + Ax) -fix) 

Ax Ax 

which reduces at the limit to (2). The same result would 

be obtained if in (1) Xwere preceded by the minus instead 

of the plus sign. 

Hence, to obtain the derivative of the sum (or difference) 
of a series of terms, some of which are constants, we simply 
take the sum (or difference) of the derivatives of all the 
terms which are functions of x, ignoring those which are 
constant. 

Thus, if y = x z + 5, — = 3 x 2 . 

dx 

Again, the derivative of 

x b — x^ + x + a — b — 8 is 5 ^ 4 — 4 x 3 + I . 

The foregoing result is sometimes expressed by regarding 
all the terms, even the constants, as functions of x, and say- 
ing that the derivative of a constant term is zero. Properly 
speaking, however, a constant has no derivative, for it is 
not a function of x. 

Examples. — Find the differential quotient of: 
1. x 2 + 2. 2. x 2 + 3 + x±. 3. x s + x 5 + 19. 

4. Prove last by general method of differentiation. 

28. If a function of x is the product of a constant by 
another function of x, i.e. if 

F{x)=K4>(x), (1) 

then F'(x)=Xct>'(x); (2) 



GENERAL THEOREMS OF DIFFERENTIATION 23 

that is, the derivative of the product of a constant by a func- 
tion is the product of the constant by the derivative of the 
function. 

Proof. — When x becomes x + Ax, (i) becomes 

F{x + Ax)=K$(x + Ax). (i)' 

Subtracting (i) from (i)' and dividing by Ax, we have 
F(x + Ax) — F(x) __J£<j>(x + Ax) — K<j>(x) 
Ax Ax 

cj>(x + Ax)— 4>(x) u 

or at the limit, F\x) = Kfiix). 

Corollary. — The derivative of mx n is m times the de- 
rivative of x n , as given in § 16. Hence, it is mnx 71 ' 1 . This 
result is so often used that it should be carefully memorized. 
When n is i, the derivative is simply m. (Show this directly, 
by § 1 8.) 

Examples. — Differentiate 

5 x s , 2 x~, 4 x 10 , $x, J x z , 

3 5 V i-v^y 

29. If a function of x is the product of two functions of 

x, i.e if F(x) == <j>(x)ij/(x), then 

F{x + Ax)= </>(•* + Ax)i//(x + Ax). 
Subtracting and dividing by Ax, we have 
F(x+ Ax) — F(x) _ $(x + Ax)i//(x + Ax) — 4>(x)if/(x) 
Ax Ax 

The right member may be changed in form without suf- 
fering any change in value by adding and subtracting 
<£ (x) \j/ (x + Ax) in its numerator, giving 
</>(x + Ax)\l/(x + Ax)-<f>(x)\l/(x)-<t>(x)\//(x + Ax)+<l>(x)\l/(x + Ax) 



24 INFINITESIMAL CALCULUS 

Grouping the terms according to common factors, we 
have 

\_^{x-\-Ax) — <t>(x)~\\\/{x + Ax) +4>(x) \\f/{x + Ax) —*p(x)"] 

Ax 
or 

LaX iJaX 

or at the limit, F\x) = $ {x)\l/(x) +ij/'(x)^(x). 

In words, the derivative of the product of two functions is 
the sum of the products obtained by multiplying the derivative 
of each function by the other function. 

Thus, ^d+^)] = ^!) (l+ , 2)+ ^+^), 2 

dx dx dx 

— 2X{\ -f X 2 ) + 2 X ' X 2 
= 2 X ( I + 2 X 2 ) . 

Examples. — 1. Find the derivative of (i + x 2 )(i — x 2 ) first by 
§ 29 and afterwards by multiplying out and then differentiating. 

2. ( 2 + x* - * 4 )(5 + * 5 )> 40* 2 + 0(* 8 -.2). 

«(3^ 2 + 4)(5^ 3 + 6^ 2 +7^ + 8), (a + b)(kx m + hx m +p)(qx 2 + r). 

3. Prove § 28 by using § 29, regarding Hsa form of ^(*), whose 
derivative is zero. (See § 27, end.) 

30. Corollary. — If F {x) = fi(x) f^x) fo(x) , we may abbrevi- 
ate fi(x)fz(jc) to (V), so that 

F( x )=fi(x) 0OO> 
whence /* (*)=/i'00 ^W+^'W/iW- 

Replacing 000 by its value ./2OO/3OO and ^' W b ^ its value 

/2 , W/sW+/3 , W/2W. 

we have 

/■'(*)=/!'(*) L/K*)yS(*)] + [/i'W/sW+Zs'W/sWl/iW 



GENERAL THEOREMS OF DIFFERENTIATION 25 

By successive applications of § 29 this theorem can be generalized 
to the product of any number of functions, and in words is as follows : 

The derivative of the product of any number of functions is the 
sum of the products obtained by multiplying the derivative of each 
function by the product of all the other functions. 

Examples. — Find the derivatives of 

(* 2 + i)C*+ 0(>- 1), * 3 (* 2 + 2* + 3)(2.*±-7)(4-* 5 ). 

31. If F(x) = -L-, then 
<t>(x) 



F{x + Ax) — F(x) _<$>{x + Ax) 4>(x) ' 
Ax Ax 

cf>(x) — <f>(x + Ax) 



Ax $(x)<f>(x+Ax) 

— 1 cf>(x+Ax)— <f>(x) 



cj>(x)<j>(x + Ax) Ax 

or at the limit F'(x)= r ~ T * <f>'(x) 

IX*)] 1 

[<t>(x)y 

That is, the derivative of the reciprocal of a function is 
minus the derivative of the function divided by the square 
of the function. 

Thus the differential quotient of — - is 

~^(3* 2 ) 







dx 


or 


-6x 
9x* 


or 


— 2 




(3*2)2 ' 


3*3 


32 


. Examples. 












1. 


Find the derivative of 














I I 




] 







X^ I + *»' I + X + X 2 



26 INFINITESIMAL CALCULUS 

2. Show by method of § 29, that if 

then jr, = *!±_=jt!± 

where the (V)'s are omitted for brevity. 

3. Prove the same theorem by applying results of §§ 29, 31, after 

throwing — in the form <p — 
xj, ^ 

33* We may interject here an application of the result of § 31 
to generalizing the theorem of § 16. The differential quotient of x n 
was there obtained only under the restriction that n be a positive 
integer. But if n be a negative integer, — m, then x n becomes — • The 
differential quotient of which is 



which reduces to — mx-™- 1 or nx n ~ l . 

That is, the restriction imposed in § 16 that n must be 
positive, may be removed. 
Examples. 
1. Differentiate x~ 2 . 2. Differentiate Sx~ 5 . 

3. Differentiate — • 4. Differentiate — -• 

x 1 8x* 

34. If we wish to differentiate the quotient of two func- 
tions as ftW , we can d this by combining the results of 

fix) j 

§§ 29 and 31, for the quotient may be written 4>(x) • — — — • 

xp(x) 

Thus, the differential quotient of x is obtained by writing it 

1 — x s 
(1 + x 2 ) — - — r - Applying the theorem for products, we get 



d(i +* 2 ) 



(i + *«) v . ; +, 

v y dx \ 1 — x 6 ) ax 

which can readily be reduced. 



GENERAL THEOREMS OF DIFFERENTIATION 27 

If the student prefers, he may simply memorize the result of example 
2, § 32, and apply. 

35. If % is a function of y, and y of x, an increment Ax 
of x produces Ay of y, which in turn produces Az of z. 

Evidently — = — • -=£- 

Ax Ay Ax 

Hence at the limit — = — • -J-- 
^r ^K dk 

This may also be expressed : 

If F(x)=<f>(f(x)), 

then F'(x)=4>'(f(x))f(x). 

It must be carefully noted that <£'(/(.*)) means the derivative" of 
<p (/(■*)), «<tf with respect to x, but with respect to fix'). It is — not 

In words, the derivative w//$ respect to x of a function of 
a function of #, is the derivative of the former function with 
respect to the latter, multiplied by the derivative of the latter 
with respect to x. 

Thus, if y =(i-f # 2 ) 3 , -2- may be found by denoting (i-\-x 2 ) by w t 
, dx d 

and then finding -2- from y = w 3 , and — from zv=i+x 2 . Whence 
,77 dw dx 

d 1= d L . ^ = 3 W z. 2x=3 ( l+x zy 2x . 

1 7 7 «-» J\ ' / 

ax aw ax • 

But the use of w is quite unnecessary, and the student should learn 
to dispense with it as well as with y also. The required derivative then 

K d(i+x 2 Y d(l+x 2 ) f , 2V2 

becomes -^ — ■ J — • -^ — J - — 2(1 + x 2 ) 2 2x. 

V(i +* 2 ) dx * K ■ ^ 

Employing the elliptical notation of differentials, the pro- 
cess is even more easily remembered and applied. The 
differential of <f>(f(x)) is 

d<t>(f(x)) or <j>\f(x))df(x), or 4>'<J(x))f(x)dx. 



28 INFINITESIMAL CALCULUS 

That is, we first differentiate treating "f(x)" as a single 
character, and our result contains df(x). We then perform 
the further differentiation indicated by this df(x). 

Thus, d(i -f * 2 ) 3 = 3(1 + x 2 ) 2 d{\ + x 1 ) 

— 3(1 + x 2 ) 2 2xdx, 
where "(1 + ^ 2 )" is first kept intact as if it were not a combination 
of symbols, but a single awkward symbol. 

36. Examples. 

1. Differentiate 4(2 + ^r 3 ) 2 . 

2. Differentiate (7 + ^) 5 . 

3. Differentiate 2(1 + 2* + x 2 ) B . 

4. Differentiate (3^ 3 — 2)" 4 . 

5. Differentiate • 

(x 2 + x + i) 2 

6. Differentiate 



(2 at 3 4- 3^ 2 + 4) 5 

7. Differentiate a + b(i + * 2 ) 2 + *(i + * 2 ) 3 + k(i + * 2 ) 5 . 

8. Differentiate 

( 3( ^ + 'fa + o 3 + , (<w . + 5 fa + , )(* - -(«* + «• + ')•). 

37* If we have 
by substituting {(*) for ^ [/(*)] and applying § 35, we have 

/*(*) = ^'RC*)]^*). 

Substituting for | its given value and for ?•' its value as obtained by 
§ 35> we have 

^'(*) = 0'OT/(*)W'[/(*)]/'G*)> 

and so on for any number of functions. If we use differentials instead 
of differential quotients, we have 

^{0i(0 2 [0 3 (--)])} = 0i , ^2 

= 0i r 02 ( 03^04 

= etc. 



GENERAL THEOREMS OF DIFFERENTIATION 29 



Examples. 

1. Find the derivative of 

4 {2(i + * 2 ) 2 + 3(1 + * 2 ) 3 } 2 + 5 {2(i + * 2 ) 2 + 3(1 + * 2 ) 3 } 3 - 

2. Differentiate {a +[b +(c + /br 2 ) 3 ] 5 } 2 . 

38. The results of this chapter may be thus summarized : 

t rf[/i(*)±/2(*)±'"]. 



dx 



=//(*) ±/«'(*) ± 



3. *^=w*). 



dx 
1 



■*'(*) 






dx 



=nA*)V'(*)- 



30 INFINITESIMAL CALCULUS 



CHAPTER III 

DIFFERENTIATION OF THE ELEMENTARY FUNCTIONS 

39. We have learned (§§ 16, 33) that the derivative of 
x n is nx n ~ Y , where n is any integer. x n is the elementary 
algebraic function. 

We have now to differentiate elementary functions called 
"transcendental." To do this we recur to the general 
method of differentiation. We first take up the trigono- 
metric functions. 

40* </(sin x) _ y sin (x + A*) — sin x 

dx Ax 

_,. sin x cos A x + cos x sin Ax — sin x 



Ax 



— lim < c 



sin Ax . I — cos Ax 1 

sin Ax 



sin x • 



I 



But becomes unity at the limit when Ax becomes zero, and 

Ax 

1 — cos Ax , 

becomes zero. 



These are shown by means of Fig. 4, where AB is an arc Ax on a 

unit radius OA. So that BC is sin A*, C6> is cosA#, and CA is 

I — cos Ax. 

^£ i s therefore *£, 
Ax BA 

j 1 — cos Ax . C4 

and ■ is • 

Ax BA 



DIFFERENTIATION OF FUNCTIONS 



31 



When BA becomes zero, CA and BC become zero. The proof 

it lim — 
B. 
ing hints : 



Fi C C 'A 

that lim = I, and lim = o, is left to the student with the follow 

BA BA 




1. I < < = , which approaches I as limit. 

BC BC CO vv 



CA CA BC BC BC 



which approaches o x I. 



BA BC BA CE BA 

u </(sin x) 

Hence — ^ J - = cos x x I — sin x X o 

dx 

= cos jr. 
In like manner, we may prove 
a 7 cos. 



dx 



■ = — sin x. 



/sin 



41. 



dtanx 
dx 



f sin x \ 
\cos x J 



dx 

cos x cos x + sin x sin x 
cos 2 .* 



Similarly, 



d(co\.x') __ — 1 
dx sin 2 x 



32 INFINITESIMAL CALCULUS 



42. 


dsec x 
dx 


\cosxJ 


— 1 — etc., accoiding to § $L f 


and 




flr(cosec^) \sin^/ 


dx dx 


43- 


d(a*) _ 
dx 


= lim aX+Ax -^ 

Ax 

-lima*.^- 1 . 

Ax 


Now let 


a* x - 1 = 5, so that a** = 1 + 5, 


and 




Ax log a = log(i + 5), 


and 




log a 


Then 




d(a*) .. 5 




a& log (1+5) 
log« 










*^^ log (1 + 5) 
5 






= lim a x log a =- 



log {(1 + 5) 5 } 

1 

The limit of (1 + 8) 8 , when 5 becomes zero (which evidently occurs 
when Ax becomes zero) is 2.718 approximately, and is called <?.* 

* This fundamental magnitude may be pictured as follows : Suppose 
interest is at 4% corresponding to "25 years purchase." $1 com- 
pounded yearly for this 25 years amounts to (1.04) 25 . Compounded 
half-yearly for the same 25 years, it is (1.02) 50 ; quarterly (1.01) 100 ; 

daily (1 + "sef 00) 5 ~5 momently, lim (1 -f- 5) , or e. Thus e is 
simply the amount of $ 1 at continuous interest during the " purchase 
period." This is $2,718, whereas with quarterly compounding the 
amount would be $ 2.705, and with yearly, $ 2.666. 



DIFFERENTIA TION OF FUNCTIONS 33 

Hence at the limit, 

d(a x ) , , I 

— ^ — L = a x log a 

dx log e 

This result is independent of the system of logarithms. It is true 
of " common logarithms." If we take e as the base {i.e. employ the 
Naperian system), then log e = I; and the result simplifies to 

d(a x ) . 

— ^ — L = a x log a. 

dx 

Henceforth we shall denote common logarithms by 
" Log " and Naperian logarithms by " log." Any other sort 
of logarithms will be denoted by " log 6 ," where the subscript 
b denotes the base of the system. 



44» We now proceed to the inverse functions of those just con- 
sidered. 

y = arc sin x, means that y is the arc whose sine is x (sometimes 
the notation sin" 1 x is used), i.e. it means the same thing as 

x — sinjy. 

From this — = cos^ 

dy 



= Vi 



— Vi — x 2 . 

But — is the reciprocal of -Z, since these expressions are the 

dy dx 

limiting values of — and — =£, which are reciprocals. 



tjence 
Or 



dy i_ 

dx V7^ 

d(s.vc sin x) i_ 

dx vr^ 



Similarly, ^( arccos ^ 



dx Vi - x 2 



34 INFINITESIMAL CALCULUS 

45 • If y — arc tan x, then x = tany. 
^ _ i 

dy cos 2 jy 
= sec 2 y 
= I + tan 2 j 
= I + x 2 . 
dy _ I 
^ I + jr' 2 
a? (arc tan.*) _ I 

a& I + x 1 

^(arc_cot_£) _ — I 
^r I + x 2 



Hence 

Or 

Similarly, 



46. If y = logx, then x = by, where b is the base of the system. 

Hence — = fr \ogi,b • 

dy \og h e 

But log 6 b = 1. 

log 6 <? 



Hence — = to 



login 
Hence ^ = lpg5£. 

d&C x 

This is independent of the particular system of logarithms. 
If b = e, then log^ = 1, and the result simplifies to 

dy I , dx 

-^- — — or dy = — 
dx x x 

47* We may now still further generalize the theorem expressed in 
§§ J 6, 33. The number n has been restricted to an integer. But if 
y = x n where n is any real number, 
then log y = n log x. 

Taking the differential of each side, 
dy dx 



DIFFERENTIA TION OF FUNCTIONS 35 

Hence ± = ?l 

dx x 



That is, the restriction of §§ 16, 33, that n must be an 
integer is now removed. It may be a fraction, an irrational 
number, or any real number whatever. 

Examples. 

1. What is the differential quotient of 

3 £ 1 3 - 5 - _! I 

X 2 , x 2 , x 2 , vx, Vx, X 3 ,~ ~? 
Vx 



2. Of Vi + x 2 , (x 2 - 1)3, ija + dVx + cxh 

48. The results of this chapter may be thus summarized : 
Direct Functions. Inverse Functions. 

d(x n ) = nx tl ~ 1 dx. 
d(mx n ) = mnx n ~ Y . 

^(sin x) = cos xdx. d(zrc sin x) 



Vi - 



</(cos x) = — sin xdx. d(arc cos x) = 



,2 
dx 



Vi — x 



,2 



</(tan jc) = — - — d(axc tan x) 



1 +** 

tfYcot x) = ^- — d (arc cot x) = — — - • 

sin- x i+xr 

Log e x 

= a x log aafc. ^(log x) = — 

■ x 



36 INFINITESIMAL CALCULUS 

No function inverse to x n (or to the more general form 
mx n ) is given, since in this case the inverse is identical in 
form with the direct function. 
i 

(Thus, if y — x n , x — y n = y m , a form identical with x n , its inverse.) 

49. Examples. 

1. Differentiate 3 sin x. 

2. Differentiate 1 — a sin x + b cos x. 

3. Differentiate 2 sin x cos x. 

4. Differentiate sin x tan x. 

5. Differentiate cot* + x 2 cos.*. 

6. Differentiate log x + tan x cos jr. 

7. Differentiate x 2 a x . 

8. Differentiate (alogx — bx 2 + ^<2 x )(i — * 3 ). 

9. Differentiate sin 3 x. 

10. Differentiate cos.* 2 . 

11. Differentiate tan (1 + x + x 2 ). 

12. Differentiate log x s H (- * tan (* + # x — arc cos 3 x). 



SUCCESSIVE DIFFERENTIATION 37 



CHAPTER IV 

SUCCESSIVE DIFFERENTIATION MAXIMA AND MINIMA 

50. The derivative of 2 x* is, as we know, 8 x s . The 
derivative of 8 x 3 is, in turn, 24 x 2 . The derivative of a 
derivative is called the second derivative of the original 
function. 

When F{x) stands for the original function, and F\x) 
for its derivative (to avoid misunderstanding we must now 
call it the first derivative), then F"(x) denotes the second 
derivative, and F'"(x) the third derivative (i.e. the deriva- 
tive of F"(x), etc. 

Again, if we use the notation -^- for the first derivative. 

dx 

if) 

the second derivative is evidently ^ L 9 which is usually 

d 2 y \dx 2 ) 

abbreviated to —^ : likewise the third or —^ — - is written 
dx- dx 

—2-. and so on to — ^, — £ etc. 

dx 3 dx 4 dx 5 

51 • Examples. 

1. What is the third derivative of x h l 

2. What are the 2d, 3d, 4th derivatives of x 2 ? 

3. Differentiate successively ax n . When, if ever, will the answers 
become zero ? What sort of a number must n be to bring about such 
a result ? 



38 INFINITESIMAL CALCULUS 

4. Differentiate successively sin x. 

5. Differentiate successively tan x. 

6. Differentiate successively a x . 

7. Differentiate successively arc sin x. 

8. Differentiate successively arc tan x. 

9. Differentiate successively log x. 

52. Just as the first derivative threw light on the problems 
of velocity, tangential slope, etc., so the second derivative 
will illuminate acceleration, curvature, etc. 

We have seen that if for a falling body s = 16 t 2 , then 

ds , , v 

— = X2 /, (i) 

dt * ' W 



whence — = 32. (2) 

ds 
We may understand this result better if we designate — 

by v, as in § 6, so that (1) becomes 

V=32t, (1)' 

and (2) ^=32, (2)' 

at 

where — is evidently simply 
dt 

d 2 s 

— 7~> 

dt 
for both are mere abbreviations of 

\dt 



dt 

What does equation (2) or (2)' mean? — means the rate 
at which the body is gaining speed. It is clear that moving 



SUCCESSIVE DIFFERENTIATION 39 

bodies do gain or lose speed, and that some gain or lose 
faster than others. 

The gain or loss of speed has nothing to do with how fast 
a body is going. A slowly moving body may be gaining 
speed very fast, while a fast moving body may not be gain- 
ing at all, or may even be losing speed. 

If we use the term veto to indicate a unit of velocity, or 
one foot per second, we know from (i) that a body which 
has fallen 2 seconds has then a speed of 64 velos, while at 
the end of 5 seconds its speed is 160 velos. Here is a gain 
of 96 velos in 3 seconds, or an average of 32 velos per 
second. 

This does not, of course, imply that the body had gained 
at the rate of 32 velos per second all the time. But equa- 
tion (2) tells us that this is the case. A falling body on the 
earth is consta?ttty gaining velocity at the rate of 32 velos 
per second. 

Rate of gain of velocity is called acceleration, and we see, 
therefore, that a falling body is a case of " uniformly accel- 
erated motion." 

Observe that the acceleration or rate of gain of velocity expressed in 
32 velos per second, cannot be expressed as any number of feet per 
second. On the contrary, substituting for the word " velos " its defi- 
nition " feet-per-second," we see that 32 velos per second is 32 feet per 
second per second. 

If the distance a body moves in time t is not 16 f 2 , but 10 1 3 , then its 
velocity is 30 r 2 , and acceleration 60 1. In other words, its acceleration 
in this case depends on the time. If the body has fallen 2 seconds, its 
acceleration is 120 velos per second ; if 3, 180 velos per second ; etc. 

53. If F(x) expresses the ordinate of any point on a 
curve when the abscissa is x, we have seen that F'(x) 
expresses the tangential slope at that point. What does 
F n (x) represent? Evidently the rate at which that slope is 



40 



INFINITESIMAL CALCULUS 



changing at that particular point as x increases. It denotes 
what we may call the curvature at that point with respect to 
the axis of x. 



Fig. 5. — A, rate of gain of slope positive ; B (" point of inflection"), zero ; 
C, negative. 

Curvature, however, is usually measured with respect to 
the tangent itself. The expression for this, the more proper 
sense of curvature, is somewhat more complicated. At a 
point when the curve is horizontal, the two sorts of curva- 
ture are identical. 



54. When the curve is horizontal, the slope of the tan- 
gent F\x) is, as has been seen, zero. But the curve may 
be horizontal at three sorts of points : a maximum as at A 




Fig. 6. — Points of zero slope: A, maximum; B, horizontal point of inflection; 
C, minimum; D, maximum. 

and D (Fig. 6), or a minimum as at C, or a horizontal point 
of inflection as at B. 

A maximum point on a curve is a point such that the 
ordinate, or y, of that point is larger than the ordinates of 
points in its neighborhood on either side. (The phrase 



SUCCESSIVE DIFEEREXTIATION 41 

" points in its neighborhood " means all points on the curve 
within some small but finite distance on either side.) A 
minimum point is one whose ordinate is less than the 
ordinates in its neighborhood on either side. A point of 
inflection is one where the neighboring parts of the curve 
on opposite sides of the point are also on opposite sides of 
the tangent as at B in Figs. 5 and 6. 

In the neighborhood at the left of a maximum the slope 
of the curve is positive, while on the right it is negative. 
For a minimum, the slope is negative on the left and positive 
on the right. For a horizontal point of inflection, the slope 
is positive on both sides or else negative on both sides. 

It is to be observed that a curve may have more than one maximum 
or minimum, and that a maximum ordinate does not mean the greatest 
ordinate of all, but only the greatest in its neighborhood. Thus the 
ordinate at D is a maximum, though that at A is larger. 

55. Dropping the symbolism of the curve, it is clear that 
when a function F(x) reaches a maximum or minimum, then 
F'(x) = o, for F'(x) represents the rate of increase of F(x), 
and at a maximum or minimum this rate is zero. 

But if, conversely, we have F'(x)=o, we simply know 
that for that particular value of x which satisfies this equa- 
tion F(x) is not increasing nor decreasing. We cannot tell 
whether it is a maximum or a minimum or an " inflectional 
stationary" value {i.e. one such that F(x) will increase for a 
change of x in one direction and decrease for a change of x 
in the other direction). 

56. Now these questions can be settled by recourse to 
the second derivative, provided this is not also zero. 

If the second derivative be positive, the function is a 
minimum ; if it be negative, it is a maximum. This will be 



42 INFINITESIMAL CALCULUS 

clear if we remind ourselves of the meaning of the second 
derivative. It indicates the rate of change of the slope. If 
positive, it means the slope is increasing ; if negative, it 
means the slope is decreasing. 

If, therefore, at a point where the first derivative or slope 
is zero, the second derivative or "curvature " (§ 53) is posi- 
tive, we know that at that point the slope is increasing. But 
as its present value is zero, it must be changing from a nega- 
tive to a positive value. This can evidently only occur at a 
minimum. Per contra, if the second derivative is negative, 
it indicates a slope growing less, i.e. (as the slope is now 
zero) changing from positive to negative. This evidently 
occurs at the maximum, and nowhere else. 

Thus, take the function x s — 27 x. This has for first derivative 
3^ 2 — 27, and for second derivative 6x. Putting the first expression 
equal to zero and solving, we find x — ± 3 ; that is, the function 
x z — 27 x has two points at which it is stationary (or the tangent is 
horizontal), where x is 3, and where x is — 3. The first of these is a 
minimum, and the second a maximum ; for the second derivative 6 x is 
positive for x — 3, and negative for x = — 3. 

57. The exceptional case mentioned in § 56 (viz. where 
the value of x, which renders the first derivative zero, also 
renders the second derivative zero) seldom occurs in 
practice. When it does occur, we cannot decide the nature 
of the function for that point, without recourse to the third 
derivative. If this be positive, the function is neither at a 
maximum nor minimum, but at a hori- 
zontal point of inflection, as at A (Fig. 
7), when, for an increase of x, the 
Fig. 7 . function was increasing, both before 

and after the point. If, on the other hand, it be negative, 
the function is at a horizontal point of inflection as at B 



SUCCESSIVE DIFFERENTIATION 43 

(Fig. 6), when the function was decreasing both before and 
after reaching this point. If, finally, it be zero, we are again 
left in the dark as to the nature of the function, and must 
proceed to the fourth derivative. We employ this just as if 
it were the second. If it turns out zero, and forces us to 
consider the fifth, we employ this just as if it were the third, 
and so on. 

That is, as long as the successive derivatives turn out zero, 
we go on until we find one which is not zero. If this deriva- 
tive be of an even order (i.e. 2d, 4th, 6th, etc., derivative), 
we know that the function is either a maximum or a mini- 
mum, and is the one or the other according as the derivative 
in question is negative or positive. But if the derivative 
which does not vanish is of an odd order (i.e. 3d, 5th, etc.), 
we know that the function is neither at a maximum or mini- 
mum value, but at a point of horizontal inflection and is 
increasing or decreasing according as the derivative is posi- 
tive or negative. 

58. We shall not devote the requisite space here to proving the 
truth of the last section in full, but shall merely indicate the first step, 
leaving the student, if he so desires, to extend the demonstration. 

Suppose in testing the function F(x) we find for the value of x 
which renders F'(x) = o, that F r, (x) is also zero, but F ,,f (x) is posi- 
tive. Denoting this value of x by x\> we may state the problem as 
follows : given 

E f (x{) = 0, 
/*'C*i)=o, 
/•'"Oi)>o, 
to discover the nature of F(x{). 

We shall solve this by reasoning from F' n successively back to F ,f t 
F\ and F. 

Since F ,n (x\) is positive, it shows that F ,f (x) is increasing as x 
increases. But as F n (x{) is zero, the fact that F n (x) is increasing 



44 INFINITESIMAL CALCULUS 

shows that it was negative before reaching F n (x\) and positive after. 
This is our conclusion for F n . 

Since F n (x) was negative before reaching F n (x{) it shows that 
F'(x) was then decreasing, and since F"(x) was positive afterward, 
F'(x) was then increasing. 

But, if F'(x) is zero at F'(xi) and was decreasing before and in- 
creasing after, it must have been positive both before and after. This 
is our conclusion for F'. Since F 1 is positive both before and after, 
it shows that F(x) was increasing both before and after, and is there- 
fore not a maximum, but a horizontal point of inflection. 

Thus let F(x) be 

x±-6x 2 + %x + 7. 

Then F' is 4.x 3 — 12^ -f- 8. 

Then F" is 12* 2 - 12. 

Then F'" is 24^. • 

The roots of F' = o are 1 and — 2. For x = I, jF" vanishes, but 
F nt is positive. Hence we know that F or x^ — 6^ 2 + Sx + 7 is at a 
stationary inflectional value increasing on either side, as ^ increases. 

But for x = — 2, F n is positive. Hence for this value of x, F is a 
minimum. 

59* Examples. — 1. Find maximum or minimum value of x 2 . 

2. Find maximum or minimum value of 3 x 2 — 27 x. 

3. Find maximum or minimum value of 2x 2 + x + I. 

4. Find maximum or minimum value oi x z — 12 x -\- 6. 

5. Find maximum or minimum value of 2x z + 6 x 2 + 6x -f 5. 

6. Find maximum or minimum value of x s — 2x + 3 jc 2 — 4. 

7. What is the nature oi x^ — 24 x 2 -\- 16 x -{- 10 for ^ = 2 ? 

8. What is the nature of ^ 4 + 4 jc 3 — 6 x 2 — 20 * -J- 17 for .* = — 1 ? 

60. If F(x) is of the form <j>(x) + X, where X is any 
constant, then the same values of x render F(x) a maxi- 
mum or minimum as render <j>(x) a maximum or minimum 
respectively. 



SUCCESSIVE DIFFERENTIATION 45 

For the nature of F{x) or of <p(x) as to maxima and minima de- 
pends exclusively on the nature of their derivatives, and the deriva- 
tives of these two functions (viz., 0(#) + K an d 000) are evidently 
identical. 

Thus to find the value of x to render 



* 2 + 2 



('^) 



a maximum or minimum, we may drop the constant term and simply 
inquire for what value of x the form x 2 is a maximum or minimum. 

6i. If F{x) is of the form K$(x) when K is a positive 
constant, then the values of x which render F (x) a maxi- 
mum or minimum are the same as those which render <jE> (x) 
a maximum or minimum respectively. 

If F(x) = K<f>(x) where K is a negative constant, then 
the values of x which render F{x) a maximum or minimum 
are the same as those which render <j>( x ) a minimum or 
maximum respectively. 

For the successive derivatives of these two functions (viz., K(f)(^x) 
and <f>(x)) are 

k$\x) ] r *'(*), 

K$"(x) \ and j 4>"(x), 
etc. J [ etc., 

and evidently the very same values of x will make the two first deriva- 
tives zero, and, if K be positive, will make the two second derivatives 
of the same sign or both zero; but if K be negative, will make them of 
the opposite sign or both zero. Similarly for the two third derivatives, 
etc. Since the natures of F and of 0, as respects maxima and minima, 
depend exclusively on the signs ( + , — , or o) of their derivatives, 
the theorem is proved. 

Thus, to obtain the value of x which will make 



i'-^Y**-* 



a maximum, we drop the constant factor (which is evidently positive) 
and find out which values of x make x 2 — x, a maximum or minimum. 



46 



INFINITESIMAL CALCULUS 



Examples. — 1. Interpret the theorems of §§ 60, 61 geometrically. 
2. Find maximum or minimum of 5(1 -j- x + * 2 )+ 10. 



3. Find maximum or minimum of — $xl x -\- 1 + : 



4. Find maximum or minimum 



62. The subject of maxima and minima is one of the 
most important in the Calculus, and has innumerable appli- 
cations in Geometry, Physics, and Economics. 




Let ABC (Fig. 8) be any triangle, and EFKH a rectangle in- 
scribed within it. This inscribed rectangle will vary in size according 
to its position. If too low and flat, it is small. If too high and thin, 
it is also small. Between these positions there must be a position of 
maximum, where the area is the largest possible. 

Now its area is the product of the base HK or EF by the altitude 
DM, and the problem consists in discovering where EF-DM is a 
maximum. 

To do this, we must first express EF and DM in terms of some one 
variable. Out of the many possible {e.g. BH, BK, AE, FC, EH, HK, 
etc.) we select AM, and denote it by x. We call AD = h and BC=a. 
Evidently AID = h — x. To express EF in terms of x, we proceed as 
follows : The triangles AEF and ABC are similar, so that their bases 
and altitudes are proportional. That is, 

AM = EF x = EF 
AD ~ BC h a' 



SUCCESSIVE DIFFERENTIATION 47 

whence EF= — 

h 

Consequently EF X DM = (k-x) — • 

h 

We wish to know for what value of x this expression is a maximum. 

We may omit the positive constant factor -, leaving 

h 

{h — x)x or hx — x 2 , 
the first differential of which is h — 2 x, 
which, put equal to zero and solved, gives 

h 

x = - t 
2 

the required answer. 

We are sure it is a maximum and not a minimum or stationary in- 
flectional value, since the second differential is — 2; i.e. negative. 

We have learned, therefore, that the maximum rectangle inscribed 
in a triangle is that whose altitude is half the altitude of the triangle. 

In physics many important principles depend upon max- 
ima and minima. Thus the equilibrium of a pool of water, a 
pendulum, a rocking chair, or a suspension bridge, is deter- 
mined by the condition that the centre of gravity in each 
case shall be at the lowest possible point. 

In economics we have the principle of maximum con- 
sumer's rent, of maximum profit under a monopoly, etc. 

63* Examples. 

1. How must a given straight line be divided so that the product 
of its two parts shall be a maximum ? 

2. "A wall 27 feet high is 64 feet from a house. Find the length 
of the shortest ladder that will reach the house if one end rests on the 
ground outside the wall." — Byerly. 

3. Find the maximum cylinder inscribed in a circular cone of revo- 
lution. 

4. Find the maximum rectangle inscribed in a semicircle. 



48 INFINITESIMAL CALCULUS 

5. A cylinder of revolution has a given diameter. What altitude 
must it have in order that it may have the least total area in propor- 
tion to its volume ? 

Hint. — Express volume and total area in terms of the variable alti- 
tude x, and the constant radius r. Then find when 
total area 



volume 



is a minimum. 



6. A house stands on a horizontal plain ; how far from it must a 
man stand in order that the length of an upper-story window may sub- 
tend the greatest angle at his eye ? The length of the window is 
given and its height from the level of the eye. 

7. If the price, /, of an article is fixed and the cost of producing it, 
for a given individual, is a function F(x), of the quantity produced 
x, how much must he produce to make his profit xp — F(x), a max- 
imum or minimum ? Express this result in words. What condition 
must F(x) satisfy that the profit may be a maximum and not a mini- 
mum ? Express this condition in words. 



TAYLOR'S THEOREM 49 



CHAPTER V 

Taylor's theorem 

64. We know that certain functions can be developed in 
terms of powers of variables. Thus {a + x)* becomes by 
the binomial theorem 

a 4 + 4 (fix + 6 a 2 x 2 + 4 ax 3 + x A . 

Again, by simple division, we may show that 

-= I — X + X 2 — X 3 + -•-. 



I -j- X 

Now the Calculus supplies a much simpler and more gen- 
eral method than algebra of developing functions in series 
of this sort. 

Thus, let cj>(x) be any function of x developable in the 
form 

<l>(x)=A + B(x- a)+ C{x - a) 2 + Dix - af + — , 
where a, A, B, C, etc., are constants, and the series con- 
verges. We shall show how to express the "undetermined 
coefficients " A, B, C, etc., in terms of the single constant a. 

By successive differentiation, we have * 

<t>'(x) = B + 2 C(x - a)+ $D{x - a) 2 + ... 

<£"(*)= +2C +2 - 7 > D(x-a)+ ->> 

etc. 

* By § 26 which can readily be extended so as to apply to an infinite 
number of terms if, as is here assumed, the sum of these terms con- 
verges, 



50 INFINITESIMAL CALCULUS 

Since these equations (and the original from which they 
are derived) are true for any value of x, they are true when 
x = a. 

They then become 

$ (a) = A, cr A= 4>(a) ; 

cf>"(a)=i.2 C, C 






3! 



etc., 
where 2 ! means 1 • 2 and 3 ! means 1 • 2 • 3, etc. 
Substituting these values of A, B, C, V, etc., we have 

<f>(x)= <f>(a)+ <!>' (a)(x - a)+ cf>" (a) ( x ~ rf 

2 ! 

3! 

65. This series, which is " Taylor's theorem," expresses 
the magnitude of the function <j> for any value of x in terms 
of its magnitude and that of its derivatives for any other 
value of x. 

Thus if we could write down some exact formula y = <f> (x) for the 
population (y) of the United States in reference to the time (x) 
elapsed since, say 1800, Taylor's Theorem tells us that we could get 
the population in 1900, <p (.*•), merely from data of the census of 1890. 

As a first approximation we take the population of 1890 itself, <p (#). 
But, as the population has not remained stationary, we add a correction 
for the increase within the decade. 

This increase we first assume to be (x — a) 0'(tf), i.e. the rate of 
increase known to exist in 1890, </>'(#), multiplied by the time between 
the two censuses (x — a). But since the rate of increase (by which is 



TAYLOR'S THEOREM 



51 



here meant so many thousand souls per year, not the percentage rate) 

<p"(a)(x - a) 2 
has not remained stationary, we add another correction — > 

constructed on the supposition that the rate of increase of the rate of 
increase of population, 0"(#), known to exist in 1890 has remained 
constant until 1900. Not content with this, we take into account the 
rate of increase of the rate of increase of the rate of increase of popu- 
lation, and so on. 

66. Geometrically, the theorem states that the ordinate 
of any point of the curve y = 4>(x) can be obtained from 
the ordinate, slope, "curvature," etc., of any other point. 




Thus, OB (Fig. 9) is x and BD, <p(x); OA is a and AC, <f>(a). 
The theorem tells us that the ordinate of the point D can be ascer- 
tained purely from the data as to the curve at C, viz. its height, the rate 
at which this height is increasing (*".*. its slope), the rate at which this 
slope is increasing (i.e. its "curvature" (§ 53)), the rate at which 
this " curvature " is increasing, etc., etc. In fact, the theorem states 
that the ordinate DB is the sum of various magnitudes: first, <p(a), 
which is represented by B8 (for this is the same as CA); secondly, 

55' 
(x — a)<p f (a), which is represented by 55' (for — is the slope of the 

C8 



52 INFINITESIMAL CALCULUS 

curve at C, and so = (t>'(a), hence 88 f = Cd x <p'(a) = (x — a) 0'(tf)) ; 

thirdly, J v , which is represented by d'8", when 8" is reached 

2 ! 
by drawing the curve Cd fl , which has the same curvature as the prin- 
cipal curve CD has at the point C, but retains that " curvature " (with 
respect to the .r-axis, see § 53) throughout; that is, we approach D by 
adding successive corrections. 5 is the position D would have had if 
the ordinate of the curve had remained unchanged from C (so that the 
curve would have followed the horizontal C5) ; d f is the position D 
would have had if the rate of increase of the ordinate, i.e. the slope 
of the curve, had remained unchanged from C (so that the curve would 
have followed Cd 1 ) ; 8 n is the position D would have taken if the rate 
of increase of the slope had remained unchanged from C (so that the 
curve would have followed C8"), etc. 

67. If we take the point E instead of C, so that a = o, 
Taylor's theorem reduces to the simple form 

cf>(x) = </>(o) + <ft f (o> + y K + y Y + etc. 

This is Maclaurin's Theorem. 

The student will observe that 0(o) is by no means itself zero. It is 
simply that particular value of 4>(x) obtained by putting x — o. Thus, 
if (x) is x s + 2x 2 + 117, 0(o) is 117. 



68. A second mode of stating Taylor's Theorem, and one 
often met with, is obtained by denoting the difference of 
abscissas x — a by /i, and replacing x by a + h (for, if 

x — a = h, x = a + h), so that 

or, changing our notation from a to #, 



2 ! 
where ^ now refers to the abscissa of C instead of that of D. 



TAYLOR'S THEOREM 53 

The student will also sometimes see the theorem expressed 
in the same form, but with y employed in place of h. 

69. There are many applications of Taylor's theorem in 
economics. Cournot in his Principes Mathhnatiques makes 
frequent use of it, as does Pareto in his Cours d'economie 
politique. 

When h is a small quantity, as in some of Cournot's cases 
of taxation, then the higher powers of h may be neglected, 
and we have the approximate formula 

${x + k) = ^(x)+h^(x). 

This is assuming that if the interval AB is very small, the . 
point 8' will coincide approximately with D. 

70. It will be observed that an hiatus was indicated 
in the demonstration of Taylor's Theorem. This means 
that it is not always possible to develop <f>(x) in the series 
proposed, and that the attempt to do so will give a diverg- 
ing or indeterminate series. 

It is impossible in so elementary a treatise as this to indi- 
cate in what cases Taylor's Theorem is applicable. The 
subject is one of great difficulty, and some of the most im- 
portant conclusions relating to it have only recently been 
discovered. 

71. To show the application of Taylor's and Maclaurin's 
theorems, let us use them to develop the function (a + x) n , 
assuming it developable. Since <$>(x) = (a +x) n , 

<t>\x) = n(a + x) n - 1 , 

cf>"(x) = n(n- i)(a + x) n ~ 2 , 

etc. 



54 INFINITESIMAL CALCULUS 

Hence <£(o)=a w , 

<l>"(o)=n(n— i)a n ~ 2 , 

etc. 
Hence 

*(*)= ^(o)+ V(o)x + ^>1^ + ... 

2 ! 

n . n-i . n(n— i)a n ~ 2 x 2 . 

2 ! 

a result which we already know by the binomial theorem. 

Again let us develop sin x, assuming it developable. 

Since <p(x) = smx 0(o)=o, 

<p f (x)=cosx <p'(o)=i, 

cp ff (x)-=— sinx cp ff (o) = o, 

0"'(*) = - cos x 0"'(o) = — I. 

etc. etc. 

Hence 

0(^) = 0(o)+0 / (o> + y \( + y ^ + ~. 

Jf 3 
= o -f- .*• + o h ••• 

^■3 ^5 j^-7 

= .* 1 ,+ •••. 

3! 5! 7! 

Again let us take 



x — a + I 



Since (f> (x) = , 0(#)=i, 

</>'(*) = - O - a + I)" 2 , 0'(«) = - 1, 

</>"(» = 2{x — a -f i)" 3 , 0"O) = 2, 

*"'(*) = -2- 3(*-*+l)-* f *'"(«) = - 3 1. 

Hence 

, / N . , N , 2(> — <z) 2 3!(^- — <2) 3 . 



TAYLOR'S THEOREM 55 

72. Among other important uses of Taylor's and Maclaurin's theo- 
rems is the evaluation of the two fundamental constants ir and e, the 

ratio of a circumference to its diameter and the base of the Naperian 

1 

system of logarithms (or limit (1 + 5) 5 when 8 approaches o). 

To solve the first problem we develop arc tan x. 

Since (.#) = arc tan x, 0(0) = 0, 

0'(*) = (i +* 2 )- 1 , 0'(o)= 1, 

*"(*) = -(1 +:*)-* 0"(o) = -i, 

£'"(*)= 2(1 +^ 2 )- 3 , 0"'(o)=2!, 

0iv(^) = - 2 • 3(1 + * 2 )" 4 , iv (o) =- 3 !. 

Hence <P(x)= 0(o) + 0'(o)* + // (o) — + ••• 

2 ! 



arc tan x = o -\- x \- 



r 2 2 ! X% 



21 3 



.st 2 . x d x* . 

= .r 1 h- 

2 3 4 

This formula holds true of all values of .*. Let x be unity, so that 
arc tan x, the arc whose tan is unity, is 45 or in length (on a circle of 
unit radius) — . We then have 





4- 1 » 


+*- 


1 4- 1 — 1 4- ... 




ir = 4[l - 


.14.1. 

2^3" 


-*+*-* + -]• 


To obtain e 


develop £*. 








?>w= ^ 




0(O)= I, 




<p'(x)= e x , 




0'(O)= I, 




0" (*)=**, 




0"(o) = I. 




etc. 




etc. 




0(*)=0(o)+0'(o)^ 


+ 0"(o)^+-. 




** = 1 -f * 


X 2 

2 ! 


* 3 
3l' 



56 INFINITESIMAL CALCULUS 

This being true for all values of x t is true for x = I. Hence 

,= i + i+JL + -L + .... 

73* Examples. 

1. Develop (a — x)~ 2 in series of ascending powers of x. 

2. Develop V 'a — x. 

3. Develop cos^r. 

4. Develop log (1 + x~). 

5. Develop a h+x . 

6. Develop arc sin x. 



x 



7. Develop 

2 +* 

8. Develop log(i + **). 

9. Develop J (e x - <r x > 

10. Develop sin 3 x. 

11. Develop <? x8ina . 



INTEGRAL CALCULUS 57 



CHAPTER VI 

INTEGRAL CALCULUS 

74. We have thus far been occupied with the derivation 
from F of F\ F", etc. But it is possible to reverse this 
process, and, given F ,n , or any other derivative, to pass back 
to F", F\ F. 

F'(x) was called the derivative of F(x) ; we now name 
F(x) the primitive of F f (x). The first process of obtaining 
F' from F is the subject matter of the differential calculus, 
of which the preceding chapters have treated. The process 
of obtaining F from F' is the subject matter of the integral 
calculus. 

75. In the differential calculus, we saw that the result of 
differentiation was expressed either in the differential quo- 
tient F'(x), or in the differential F\x)dx. In the integral 
calculus it is customary to employ only the latter form. We 
called F\x)dx the differential of F(x) ; we now call F(x) 
the integral of F'(x)dx. We obtained F'(x)dx from F(x) 
by differentiation. We obtain F(x) from F'(x)dx by inte- 
gration. The symbol of differentiation was d; that of in- 
tegration is J . 

Knowing that d(x 2 ) = 2 x dx, we may write I 2 x dx = x 2 ; 
or again, since 

dF (x) = F \x) dx 



58 INFINITESIMAL CALCULUS 

expresses in the most general manner the process of the 
differential calculus, 



CF'(x)dx = F(x) 



expresses the process of the integral calculus. Both equa- 
tions state the same fact looked at from opposite directions. 
The former equation reads, " the differential of F(x) is 
F'(x)dx"; the latter may be read, " the function-of-which- 
the-dirTerential-is F\x)dx is F(x)" for the hyphened words 
are what is meant by " integral of." 

The simplest form of the above equation is \ dx — x. 

76. The symbol I was originally a long S, which was the old 

symbol for " sum of" (to-day it is usual to employ the Greek 2 instead). 
Integration was looked upon as summation, dy being the limit of 
A_y, and Ay being a small part of y, the differential dy was conceived of 
as an infinitesimal part of y. An infinite number of dy's were thought 
of as making up the y. 

77. As d(x 3 ) = 3 x 2 dx, it follows that 

I 3 x 2 dx = x s . 
But d(x 3 + 5) — 3 x 2 dx \ 

hence I 3 x 2 dx = x s + 5 ; 

that is, the integral of 3 x 2 dx (or the primitive of 3 x 2 ) may be 
x 3 or x s + 5, and evidently also x' 6 + 1 7 or x 3 + any constant 

whatever. In general, 1 F\x)dx is F(x) + C, where C is 
any arbitrary constant. For the latter expression differenti- 
ated gives the former (§ 27). 

An arbitrary constant (usually denoted by C) must there- 



INTEGRAL CALCULUS 59 

fore always be supplied after integrating any differential to 
obtain the complete integral. 

78. There is no general method of integration known 
corresponding to the general method of differentiation of 
Chapter I. The only way we arrive at the primitive of a 
given function is through our previous knowledge of what 
function differentiated will yield the given function. 

79. ( ax n dx = ax + C, 
J n + 1 

ax n+1 
provided n is not = — 1. For the differential of — h C 

71 + I 

is evidently ax n dx provided n + 1 is not zero ; i.e. provided 
n is not = — 1 . 

The rule, therefore, for integrating the simplest algebraic 
function is to increase the exponent by one, and divide the 
coefficient by the exponent so increased (and then, of 
course, to add an arbitrary constant). 



Thus, 


1 2 x' 2 dx is | x z + C. 


80. Examples. 






\ 2 x dx = ? 




jW=? 




\ 3 x b dx — ? 




Cx dx _ -j 



Jdx_ ? 
x*~' 
C±dx = ? 
J x* 



60 INFINITESIMAL CALCULUS 

81. It may seem at first that a result involving an arbitrary 
constant can be of little use. But this is far from true. 
Though we cannot determine the arbitrary constant from the 
given differential, we may have, in any particular problem, 
information from some other source which will enable us to 
determine it, and often, as we shall see, we do not need 
to determine it at all. We may interpret the constant C 
geometrically by plotting the equation y = F(x) + C. To 
know F'(x)dx or F\x) is to know the slope of the curve 
for any value of x. But evidently the slope of the curve 
does not determine the curve ; since, if the curve were 
shoved up or down without change of form, it would have 
just the same slope for the same value of x. The constant 
C has to do with the vertical position of the curve. It has 
nothing to do with its form. 

82. We may profitably follow the plan adopted in intro- 
ducing the differential calculus, and begin by considering a 
mechanical and a geometrical application. 

We have seen that, knowing a body falls according to the 
law s=i6t\ (1) 

we can show that its velocity at any point is 

ds / x 

J t =Z2t. (.) 

Suppose, however, we only know that a body acquired 
velocity according to law (2), can we pass back to law (1)? 
As has been said, in the integral calculus it is customary to 
use the differential form to start with. Accordingly, we 

write (2) in the form 

ds = 32 t dt. 
Integrating, we have 



= I 32 tdt- 



32/ 



2 



+ C=i6t2+C. (3) 



INTEGRAL CALCULUS 61 

Now, although equation (2) with which we started does 
not enable us to judge of the value of C, we may evaluate C 
from outside data. 

Thus if we know that s is measured from the point at 
which the body started to fall, we know that when / was zero, 
s must have been zero too. 

Putting s = o and t = o in (3), we have 

o = o + C, 

or C = o. 

After substituting this value of C in (3), the equation 
takes the definite form 

s= 16 t\ 

83* Of course, C is not always zero. In fact, in the above ex- 
ample, we might reckon the distance s of the falling body not from 
the point where it started, but from a point 27 feet above. We then 
know that when 

/ = o, s = 27. 

Substituting in (3), we have 

27 = o + C or C— 27, 
and (3) now becomes 

s — 16 t 2 + 27. 

Evidently the value of C depends solely on what origin we use to 
measure s from. 

84. Similarly, if we know the relation between the slope 

of a curve -~ and its abscissa, we can obtain the equation 

of the curve, except for an arbitrary constant which regu- 
lates the vertical position of the curve. This example is the 
true inverse of the geometrical illustration in the differential 
calculus (§ 12). But for the purpose of the integral calculus 
we prefer another geometrical example. 



62 



INFINITESIMAL CALCULUS 



85. Suppose we have (Fig. 10) a plot of y =f(x). Give 
to x an increment Ax, viz. AE or BK, and consider the 
resulting increment not of y, but of the area OABC or z. 




This increment Az of the area is evidently the small area 
ABDE. This small area is the sum of the rectangle 
ABKE and the tiny triangle BDK. The area of the rec- 
tangle is the product of its base Ax by its altitude fix). 

So that 

Az = f{x) Ax + BDK. ( 1 ) 

Evidently the smaller we make Ax, the smaller the area 
of BDK becomes relatively to the small rectangle, and may 
finally be neglected, giving the important equation 

dz=f(x)dx. (2) 

This is not, of course, a mere approximation. It is abso- 
lutely exact. 



INTEGRAL CALCULUS 63 

The reasoning just given is to be understood as an elliptical form 
of the following: 

Dividing ( I ) by Ax, we have 

Az r, n , BDK , . 

^ = /W + ^T (3) 

Now is less than 

Ax 

rectl/K . rectl/K 
; i.e. 

Ax BK 

But the area of a rectangle divided by its base is its altitude — in 
this case DK. Hence (3) may be written 

— =f(x) 4- something less than DK. 
Ax 

It is evident that when Ax becomes zero, DK becomes zero, and 
" something less than DK becomes zero," so that our equation becomes 

£=/(*). 

ax 
which may be written 

dz = f(x)dx. 

This equation is often written 

dz = y dx, or z = \ y dx 9 

y being the usual symbol for f(x), the ordinate of a curve. 

86. Suppose y or f{x) to be 

3 x ~ + 5 i 
that is, let y = 3 x 2 + 5 be the equation of a curve. The 
integral calculus enables us to obtain the area z in terms of 
the abscissa x. 

We know that dz = (3 x 2 + 5) dx, 

Z =J (3^ 2 +S)^ 

z = x s + 5 x + C. (1) 



64 INFINITESIMAL CALCULUS 

The student may test the correctness of this integral by 
differentiating it and obtaining (3 x 2 + $)dx. 

It remains to determine C. Since we intended to meas- 
ure the area z from the jy-axis, evidently z vanishes when x 
vanishes. Putting x and z both equal to zero in (1), we 
obtain C = o. (If we had measured area from some other 
vertical than the jy-axis, the value of C would be different.) 
Hence (1) becomes z — x 3 + 5 x. 

Thus suppose x = 3; then z = 42. That is, the area included 
between the curve y — $x 2 + 5, the axes of coordinates and a vertical 
3 units from the jj/-axis is 42 units. If the linear units be inches, the 
area units are square inches. 

87* We see more clearly now than in § 76 why integration was first 
conceived of as summation. The area z is evidently the sum of a great 
many Az's, and at the limit is conceived of as the sum of an indefinite 
number of dz's. 

The dz, though a real zero, is thought of as an elementary strip of 
area infinitely narrow — the limit of ABDE. 

88. The problem of obtaining curvilinear areas was one of the 
earliest and is one of the most important of the applications of the 
integral calculus. Previous to the discovery of this branch of mathe- 
matics only a very few curves, such as the circle and parabola, could 
be so treated. 

89. We are here chiefly interested in the geometrical 
symbolism. We have seen that the slope of a curve is 
the differential quotient of its ordinate (with respect to its 
abscissa). We now see that the ordinate in turn is the 
differential quotient of its area (also with respect to the 
abscissa). For dz=ydx means simply 

dz __ 
dx 



INTEGRAL CALCULUS 65 

If we wish to make a graphic picture of any function and 
its derivative, we can represent the function either by the 
ordinate y of a curve or by its area, while its derivative will 
then be represented by its slope or ordinate respectively. 

If we are most interested in the function, we usually 
employ the former method (in which the ordinate repre- 
sents the function) ; if in its derivative, the latter (in which 
the ordinate represents the derivative). That is, we usually 
like to use the ordinate to represent the main variable under 
consideration. 

Jevons in his Theory of Political Economy used the 
abscissa x to represent commodity, and the area z to repre- 
sent its total utility, so that its ordinate y represented- 
"marginal utility" (i.e. the differential quotient of total 
utility with reference to commodity). Auspitz and Lieben, 
on the other hand, in their Untersuchungen ilber die Theorie 
des Preises, represent total utility by the ordinate and margi- 
nal utility by the slope of their curve. 

90. The method of integration enables us not only to 
obtain the particular curvilinear area described, but also an 
area between two limits, as AB and A f B' (Fig. 10). Evi- 
dently this area is the difference of two areas OAB'C 

and OABC. The first is the value of I f(x)dx, when 

OA ] (or x 2 ) is put for x in the integral when found, while 
the second is the value of the same integral for x = OA 
(or Xi). This is expressed as follows : 



X36=X 2 
f{x)dx, 



and is called an integral between limits, or a definite integral. 
The reason it is called definite is that it contains no arbi- 



66 INFINITESIMAL CALCULUS 

trary constant, for this constant disappears when one of the 
two integrals concerned is subtracted from the other. 

Thus, if \f(x)dx be F(x) + C, 

J^x=x 2 
f(x)dx 
x=x ± 

means simply (F(x 2 ) + C) - (F(x x ) + C), 

which reduces to F(x 2 )—F(x 1 ), for C must be taken to be 
the same in both integrals. 

The area between the curve $x 2 -f- 5, the x axis, and the two verti- 
cals erected at x = 2 and x = 4 is 

f *"* (3 * 2 + 5)*k = I> 3 + 5 * + C] x=4 - O 3 + 5 x + C]*=2 = 66, 

Jx—2 

for the C drops out, since for each expression the area is measured from 
the same vertical, though no matter what vertical. 

It is usual to abbreviate the expression for limits. 
f(x)dx, we write I f(x)dx. 

x=2 t/2 

91. There are certain general theorems of integration 
corresponding to the general theorems of differentiation of 
Chapter II. Of these the two most important are : 

Cx/(x)dx =KCf(x)dx 

and f[/i(x)±/ 2 (x)± -)>** 

= JA(x)dx ±j f 2 (x)dx ± j f 8 (x)d: 



f x ±. 



The proof of the first is simple, for the integral of the 
right side of the proposed equation is K(F(x) + C), or 
KF(x) + KC or XF(x)+C, where F(x) means the primi- 



INTEGRAL CALCULUS 67 

tive of f{x) and C is an arbitrary constant. But C might 
as well be written C, since its value is anything we please. 

The integral on the left is also KFix) + C ; for this 
differentiated gives Kf(x)dx. 

The proof of the second is also simple. If we denote 
the primitives of fi(x), f 2 (x), •••, by F^x), F 2 (x), •••, it is 
evident that the integral on the right is 

Fi(x) + d ± F 2 (x)+C 2 ± F 3 (x) + C 3 ± »., 
or F 1 (x)±F 2 (x)± ••• +C, (i) 

where C is C\ -f- C 2 + C 3 , and is therefore arbitrary. The 
integral on the left is the same quantity (i), for the differ- 
ential of (i) is (§ 26), 

diF^x) ± F 2 (x) - • + C)- dF^x) ± dF 2 (x) - - - 
=fi(x)dx ±f 2 {x)dx ... =(fi(x)±f 2 (x) ... )dx. 

92. Examples. 

1. Integrate (1 + a + b)x 2 dx. 

2. Integrate x 2 dx -f 7 -y 3 d£t + 5 ^ 5 dx. 

3. Integrate (/* -f 2){cx* dx + £* 6 afcr}. 

4. If the velocity of a body increases with the time according to 

the formula — = 3 t 2 . find the formula for the distance traversed. 
dt J 

5. How far does it move between the instant when t is 3 seconds 
and that when t is 5 seconds ? 

6. Find the expression for the area (corresponding to z in Fig. 10) 
for the curve whose equation is y — 5 x 2 -f 2. 

7. What is the value of that area for the point where x is 1 ? 
'Where x is 3 ? Where y is 22 ? 

8. What is the area between the curve, the ^f-axis, and the two 
verticals erected at x = 2 and x = 4 ? 

9. Solve the same problems for the curve y = x s + 14 ; for _y = 
x 2 ; for/ 2 = 4d\r. 

10. Find the area z, for y = a x ; j = log (•*" + 5) ; JK = sin^r. 



68 INFINITESIMAL CALCULUS 

93. Just as we may differentiate successively, so we may 
integrate successively. 

If we perform the integration 

J f(x)dx and obtain fi(x), 

we may then take 

I f Y (oc)dx and obtain f 2 (x), 

and then I f 2 (x)dx and obtain f 3 (x), 

etc. etc. 

Instead of writing I / 1 (x)dx, we may substitute for f\{x) 
its value J f(x)dx y and we shall have 

I \ f(x)dx Idx, 

which, however, is usually abbreviated to I I f{x)dx dx, or 
even to I i f(x)dx 2 . 

Similarly, we may write 

I I lf(x)dxdxdx, or I l if(x)dx s , etc. 

We may express the double, triple, etc., definite integrals 
also. The full form for the double definite integral would be 



Jr»x=bf fx=k 
f(x)dx 
x=a [yx= " 



which, however, may be condensed to 



dx 



£j>> 



dx 2 , 



INTEGRAL CALCULUS 69 



94* To apply these ideas we recur to our old example of a falling 

ds 
body. Suppose our first knowledge is not s = 16 1 2 nor — = 32/, but 
,.-> dt 

— = 32; that is, we simply know that the acceleration is a given con- 
dt 2 

stant (32 velos per sec), or to be more general let us call this con- 
stant g. , I ds 



The given equation, — - = g, means, as we know, — - — L — ? or 
dt' 1 dt 

whence, integrating, — —gt-^C; (1) 

dt 

but this may be written ds = gtdt + Cdt, 

whence, integrating again, s = \gt 2 + Ct + K. (2) 

We have still to determine the arbitrary constants C and K. If the ■ 

distance s is measured from the starting-point then, s and t vanish 

simultaneously. Substituting zero for them both in (2), we obtain 

K=o. 

It remains to determine C. 

To do this we take equation (1) and suppose the body falls, not 

from rest, but with an initial velocity of 11 feet per second; then when 

ds . 
t is zero, — is u, 

dt 

and (1) then reduces to 

u — o + C or C = u. 
Substituting C '= u and K — o in equation (2), we have 
s = igfi + ut, 
the general equation of falling bodies. 

95* The process which we have followed out in detail from the 

equation 

d 2 s 

raav be condensed as follows : 






= % g p + a+K. 



70 INFINITESIMAL CALCULUS 

96. The simple transcendental integrals are obtained as follows : 
Since <^(sin^)= cos^^r, then I ao^xdx = sin^r + C. 

Since d(cos x) = — sin x dx, then 1 — sin x dx — cos x -f- C, 
whence I sin (jx)dx = — cos^ — C = — cos.* -f C, 

for C is perfectly arbitrary. 

c . v t n a x hogadx ,, Ca x Logadx „ . „ 

Since d{a x ) — 2 , then I § = a x -{- C, 

Log ^ J Log e 

whence I a x dx— s_ _f- (7. 

J Log a 

Also (Vdfr=-^— + C. 

J log^ 

Since </arc sin.r = — x then 1 — ==== = arc sin^r + 6". 

Vi — # 2 J Vi — x 2 

Since (^arc tanx == — — , then \ = arc tan x -f C. 

1 + x 2 Ji+1 2 

n oc 1 doc 

Since dlogx = — , then I — = logx+C 

x J x 

= \ogx + log K = log (JCx) = log ( Cx), 
lor C and K are wholly arbitrary. 

97. We may summarize the formulae for integration which 
have been given : 

I adx = ax + C, 

/x n+1 
x n dx = h C (when n is not = — 1), 
n+ 1 

I jr -1 dfr = log x + C, 

J Log a 

= ^ + c, 

log a 



INTEGRAL CALCULUS 71 



Ce x dx = e*+ C, 



/; 



arc tsinx + C, 



f-1 

J Vi 



+ * 2 

dx 



= arc sin ^ + C, 



i sinx dx = — cos # + C, 
I cos .r dfc = sin # + C. 



98. Treatises on the integral calculus are usually very bulky, be- 
cause they are occupied with the determination of special integrals, 
both definite and indefinite, and with special devices for obtaining 
them. In this little book, which is devoted to only the most general 
and fundamental principles, we may fitly close our discussion at this 
point. Practically, even advanced students of the Calculus usually 
depend on tables of integrals. The reader is referred to B. O. Pierce's 
" Short Table of Integrals." Completer tables occupy large quarto 
volumes. An absolutely complete table does not exist, for there are 
multitudes of integrals which have never yet been solved. 

99. We may, however, point out one tool for integrating 
already in the reader's possession. 

Suppose we have to integrate 

x (x 2 + 2) z dx. 
This may evidently be put in the form 

(x 2 + 2) Z X dx, 

or ^ (x 2 + 2) 3 2 x dx, 

or ±(x 2 + 2 ) 3 d(x 2 ), 

or ±(x 2 + 2 ) 3 d(x 2 +2), 

and in this form it is easily integrated. 



72 INFINITESIMAL CALCULUS 

For, putting u = x 2 + 2, we iiave 

^ u 3 du, 
the integral of which is 

8 

This device consists in changing the variable, getting rid 
of dx, and obtaining instead a differential of some other 
variable, u, in terms of which the whole expression may be 
written. 



100. Examples. 
1 
2. 



x s dx 



. (\/xdx=? 7# C dx 

J J a + x 

z Cadx = ? r 2 bxdx 



2 ^r 



APPENDIX 73 



APPENDIX 

FUNCTIONS OF MORE THAN ONE VARIABLE 

101. We have had to do hitherto with functions of only 
one variable, such as x 2 + 2 x + 3- But the magnitude 
x 2 + 2 xy + 3 r, for instance, is dependent for its value on 
two variables, x and y ; i.e. is a function of x and y. 

The relation 2 = x 2 -\- 2 xy + 3 j' 2 , or, more generally, 
2 = F(x, _>'), states that 2 is a function of x and 7 ; that is, 
that a change either in x or y produces a change in z. 

Thus, the speed of a sailing vessel is a function of the strength of 
the wind and the angle at which she sails to the wind. 

The force which produces tides is a function of the earth's distance 
from the moon and its distance from the sun. 

The price of stocks is a function of the rate of dividends and of the 
rate of interest. 

Similarly, w = F(x, y, z) expresses the fact that w de- 
pends on x, y, and z, and so on for any number of variables. 

Thus, the force which guides the moon is a function of its distance 
from the earth, its distance from the sun, and the angle between the 
directions of these two distances. 

The price of a Turkish rug is a function of the prices of its constitu- 
ents, the cost of transportation, the rate of tariff, etc. 

If for w = F{x, y, z), the condition of some special 
problem should require z to remain constant, the function 
may be written asw = <£(#, y) ; and if y is also constant, as 
w = {//(x). 



74 INFINITESIMAL CALCULUS 

Thus, the speed of a sailing vessel is a function of her angle to the 
wind, if the strength of the wind remain constant. 

The price of woollen cloth is a function of the price of wool, if the 
cost of labor, etc., remain constant. 

102, Since the terms of an equation can be transposed, 
it is always possible to gather them all on the left side, thus 
reducing the right side to zero, y = -\/x 2 + i is the same 
equation as y 2 — x 2 — i = o. The left member is here a 
function of x and y. And in general it is evident that any 
relation between two variables y = F(x) can be reduced to 
the form <j>(x,y)=o. When expressed in the first form, y 
is called an explicit function of x. In the latter it is an 
implicit function of x. 

In like manner, any relation z=F(x, y) can be reduced 
to the form <£>(x, y, z) — o; any relation w = F(x, y, z) to 
<$>(x, y, z, w) = o, and so on. 

103. We have seen that <f>(x,y)=o or y=F(x) can 
always be represented by a curve with x and y as the two 
coordinates. So, also, $(x, y, z)=o or z = F(x, y) can 
always be represented by a surface with x, y, and z as the 
three coordinates. 

Draw three axes at right angles to each other, such as the 
three edges of a room, meeting at a corner on the floor, the 
.x-axis being directed, say, easterly, the j-axis northerly, and 
the 2-axis upward. 

To represent z — x 2 + 2 xy + $y 2 , 

let x have any particular value, such as 2, and y, 1. 

Then s=2 2 +2X2Xi + 3Xi 2 =n. 

Find the point in the room which is 2 units east of the 
corner, 1 unit north of it, and n units above it. This is 



APPENDIX 75 

one point of the required surface. By taking all possible 
combinations of values of x and y, and finding the result- 
ing values of z, we can find all points on the surface. 

104. When z = F(x, y), we may vary x by Ax, while y 
re?nai?is constant, and thus cause in z an increment denoted 
by Az. The ultimate ratio of Az to Ax is expressed by 

dz dF(x,y) 

7T or V - ^ 

ox ox 

and is called the partial derivative of F(x,y) with respect 
to x. 

Similarly, — or \_l21 

dy dy 

is the partial derivative with respect to y ; i.e. the derivative 
obtained by keeping x constant during the differentiation. 

Observe that the symbol d, denoting partial differentia- 
tion, is not identical with d. 

105. The geometrical interpretation of these partial deriv- 
atives can be made evident. If on the surface,-^=./ r (..r, y), 
say the surface of a stiff felt hat, we take any given point P 
and pass through it a vertical east and west plane, the plane 
and surface intersect in a curve passing through P. The 
tangential slope of this curve at P (or, as we may call it, the 

dz 
E-W slope of the surface itself) is For the coordi- 

dx 

nates of P are x, y, z, and those of a neighboring point Q 
on the curve (and therefore on the surface) are x -f Ax, y, 
z + Az, where A^r is the difference between the x's of P 
and Q, and Az the difference between the s's ; the /s are by 
hypothesis the same. The slope of the line joining />and Q is 



76 INFINITESIMAL CALCULUS 

\<7> Az c)z 

— , and its limiting value, Km — or — , is the slope of 

Ax Ax dx 

the curve at P (see § 12); i.e. the E-W slope of the sur- 
face. 

CJ Z (j fix OC V 1 

Similarly, — , or — v 'J' , is the north and south slope of 
oy oy 

the surface. 

These two primary slopes of the surface can be repre- 
sented by placing two straight wires or knitting needles 
tangent to the hat at the point P, one in an E-W vertical 
plane and the other in a N-S vertical plane. 

If we take any neighboring point R on the surface, its 

coordinates are x + Ax, y + Ay, z + Az, where the A's are 

the differences of coordinates of P and R. 

Az 
Join P and R. Then — represents, not the true slope 
Ax 

of the line PR, but its east and west slope. It is the rate 
the line ascends in comparison, not with its true horizontal 
progress, but with its eastward progress. A climber as- 
cending a northeasterly ridge may be rising 5 feet for every 
3 of horizontal progress, but yet rising 5 feet for every 2 of 
eastward progress. We have to do with the latter rate, not 
the former. 

Az 
So also — is the north and south slope of the same line PR. 

Ay 

Now let R approach P (along any route whatever upon 

the surface) until it coincides. The line PR approaches a 

limiting position which is a new tangent to the surface (a 

tangent to that curve in the surface which R traced in ap- 

Az 
proaching P). The E-W slope of this tangent is lim — , 

called — , and its N-S slope, — 
dx dy 

Representing this tangent by a third wire, we have three 



APPENDIX 77 

tangent wires through P 9 one in an E-W vertical plane, a 
second in a N-S vertical plane, and the third, any other 
tangent. The first has no N-S slope ; its E-W slope is 

— . The second has no E-W slope; its N-S slope is — 

dx . dz dz dy 

The third has both kinds of slope, viz., — and — 

dx dy 

1 06. As will be shown, the relation between these various 
derivatives is 

dz = -^- dx + ^- dy, (1) 

ox dy 

which is to be considered an elliptical form of 

dz _ dz . dz dy 
dx dx dy dx 



or of 



dz __ dz dx dz 
dy dx dy by) 



(2) 



The form (1) has the great advantage of symmetry. It 

dv dx 
seems, however, to conceal the existence of -+- or — , which 

dx dy 

are brought out in (2). These last two magnitudes require 

merely a word of explanation. -2- is not an upward slope 

dx 

at all, as it does not involve the vertical z. It is the incli- 
nation of the third wire across the floor, the rate at which 
a moving point on it proceeds north in relation to its east- 
ward progress. 

IO7. The proof of the formula stated in the last section is as 
follows : * 

* In order to master and remember this proof, the student is advised 
to construct for it some actual physical model. He will then find it 
extremely simple. 



78 INFINITESIMAL CALCULUS 

We first assume that all wires through P tangent to the surface lie 
in one and the same plane called the tangent plane. This assumption 
is analogous to that in § 14, that the progressive and regressive tan- 
gents coincide. There is an exception if the surface has an edge or 
wrinkle at the given point. 

Let us take in this plane the three tangent wires above considered, 
viz. the two primary wires (in vertical planes running E-W and N-S 
respectively) and the wire obtained as the limiting position of PQ. 
Take a point Q' on this third or " general " wire, having coordinates 
x -f A f x, y -f A'y, z + A r z. (The primes serve to distinguish Q' on 
the tangent plane from Q on the surface.) 

Through Q' pass two vertical planes running E-W and N-S respec- 
tively. We already have two such planes through P. These four 
vertical planes cut the tangent plane in a parallelogram, of which PQ' 
is a diagonal and the " primary wires " are the two sides meeting at 
P. Denote the two vertices as yet unlettered by ^Tand K, the former 
being in the E-W and the latter in the N-S primary wire. 

A'z being the difference in level of P and Q' is the sum of the dif- 
ference in level of P and H and of H and Q\ just as the difference in 
level between Mount Blanc and the sea is the sum of the elevation 
of Lake Lucerne above the sea and of Mount Blanc above the Lake. 
(It does not matter whether H is or is not intermediate in level between 
P and Q 1 , for if not, one of the heights considered becomes negative.) 

Now the difference in level of P and H is 

^-A'x, 

dx 
for the difference of level, k, between any two points, as M and N 




(Fig. 11) is the product of the slope of MN by the horizontal interval, 
a, between them (since : slope of MN = - , whence h = a x slope of 



APPENDIX 79 

MN). — is known to be the slope of PQ 1 , and A'x is the E-W 

bx 
interval between P and Q' , and therefore also the E-W interval (or 
in this case the horizontal interval) between P and H (since J7 and Q' 
are in the same N-S plane). 

Again the difference in level between //"and Q' is 

^A'y. 

by 

bz 
For — , being the slope of PIT, is also the slope of HQ' parallel to 

by 

PIC, and A'y, being the N-S interval between P and Q', is also the 

N-S (and in this case horizontal) interval between H and Q' (since 

H and P are in the same E-W plane). 

Therefore, 



This may be written 



A'z=&-A f x + &-A'y. (I)*, 

dx by 

A'z _ bz | bz t A f y ,y 

a'x bx by a'x 

Now is the E-W slope of the " general tangent" wire PQ'. 

A ' x dz A'v 

But we have seen that — is also this slope. Again, — ■- is the inclina- 

dx A'x 

tion of this same wire across the floor (the rate at which a point 
moving on the wire proceeds northward relatively to its eastward 

progress). But so also is -^ (§ io6)« Substituting therefore these 

dx 
values for the primed expressions, we have 

dz__bz_bz m dy^ 
dx b x by dx 

which may be thrown into the elliptical form 

bx by 

In this, dz is called the total differential of 0, while — dx and — dy 

bx bv 

are its partial differentials. J 

It is evident that we should reach the same result if in the preced- 
ing reasoning we had employed K in the way we did employ H, and 



80 INFINITESIMAL CALCULUS 

vice versa ; also that we could have divided (i) r by Ay instead of 
by A ! x. 

1 08. The formula (1) (§ 106), or its two alternative 
forms (2), enable us to ascertain the direction of any tan- 
gent line to a surface. 

Thus, let the surface be 

z — x 2 + 2 xy + 3 y 2 , 
and let it be required to determine any tangent line at the point whose 
x and y are I and 1 respectively; z is evidently 6. 

1. The primary E-W tangent wire at this point has an E-W slope 

— = 2x + zy = 4> found by differentiating the above equation treat- 
Sx 

ing y as constant, and has no N-S slope. 

2. The primary N-S tangent wire at this point has a N-S slope 

— = 2 x + 6y = 8, and has no E-W slope. 
by 

3. The tangent wire in the vertical plane running northeast and 
southwest has an E-W slope of 

<h__bz_ \dz_ dy_ 
dx Qx dy dx 

= 4 + 8^ 
dx 



and a N-S slope of 



= 4+8x 1 = 12, 
dz _ Qz dx Sz 



dy fix dy dy 
= 4 X I + 8 = 12. 

4. The tangent wire in the vertical plane running northwest and 
southeast has the two slopes 

4 +8(- i) = -4 
and 4(-i)+8=+4. 

5. The tangent wire in the vertical plane cutting between north and 
east so as to be advancing north twice as fast as east 



(i.e. so that ^ = 2), 



APPENDIX 81 

i i r dz d% , d% dy 

has slopes of — = ^— + ~- • - z - 

^r $.*■ oy "•*" 

= 4 + 8 x 2 = 20, 

, dz _Qz dx d% 

dy $x dy dy 
- 4 x I + 8 = io, 
and so on for any tangent wire whatever. 

109. Examples. 

1. Find the slopes of the five sorts above indicated for the same 
surface at the point for which x = 3 and y ±= 2. 

2. At the point where x — — 1, jy = — I. 

3. At the point where x — o, y — o. 

4. For the surface 2 = ^r 3 + x 2 -f # -f- xy -f j^ + > /2 + ^ 3 at the 
point x = o, y-= i. 

5. For the surface 

2 = jr 2 jj/ — 2 ^r 2 y 2 + 3 

at the point .# = 2, jy = 3. 

6. On the same surface at the same point, what are the E-W and 
N-S slopes of the tangent line which progresses northward 3 times 
as fast as eastward ? 4 times ? 3^ times ? 

7. Answer the same questions for z = logjy + 3 X + xy. 

no. When we have a function of more than two vari- 
ables, as y = F(x, y, z), there is no mode of geometrical 
interpretation corresponding to the curve for y = F(x) and 
surface for z = F(x, y) (unless, indeed, we posit a "fourth 
dimension," and speak of a " curved space " of three dimen- 
sions whose coordinates are x, y, z, w !). 

It may be shown, however, in a manner strictly analogous 
to the process of § 107, but without employing the geomet- 
rical image, that 

du/ = ^ dx + §^ dy+ §E dz . 
dx dy dz 



82 INFINITESIMAL CALCULUS 

This differential equation is elliptical for the three equations 
obtained by dividing through by dx, dy, and dz. 

The theorem and its proof are extensible to any number 
of variables. 

ill. A very important application of the principle of 
partial derivatives occurs when y is an implicit function of 
x ; i.e. when <f>(x f y) = o. We are enabled to obtain the 

derivative -2- without being obliged first to transform the 
dx 

implicit function into the explicit form y == F{x). 

Thus, if x 2 -\- y 2 = 25, we may find -3- without changing the equa- 
tion to the form y = ± V 25 — x 2 . 

112. We know from § 106 (2) that if = <j>(x,y), then 

dz^ d<j>(x,y) d<j>(x,y) ^ dy^ 
dx dx by dx 

which may also be written in two other forms, as given in 
§ 106. 

When z is zero, as in the case now being considered, then 

— is also zero (§ 27, end). Making this substitution in the 

dx 

above equation, we obtain 

d<K-*> y) 

dy dx 

dx~ dcf>(x,y) ' 
dy 

In words : To find the differential quotient of y with re- 
spect to x when the functional dependence between x and y is 
expressed in the implicit form <l>(x,y) = o, differentiate the 
function <\>(x, y) with respect to x, treating y as constant, 
and then again with respect to y, treating x as constant. 



APPENDIX 83 

Take the partial derivative found from the first differentia- 
tion, divide it by that found from the second, and prefix the 
minus sign. 

Thus, if x 2 +y 2 = 25, or x 2 + y 2 — 25 — o, we may find — as 
follows : 

The partial derivative of x 2 + y 2 — 25 with respect to x is 2 x, and 
with respect toy, 2y. Hence 

dy _ 2 x _ # 

dx 2 y y 

This result is expressed in terms of both x and y, but it may be 
transformed so as to involve but one variable. Thus, substitute for y its 
value as obtained from x 2 -{■ y 2 = 25, viz. ± V25 — x 2 . Then 

dy _^_ x 

dx ± V 25 — x 2 

a result identical with that obtained by differentiating the explicit form 



y = ±V2~f 



113. Examples. 



1. Find -*-. if xy = 1. 

dx 7 

2. Find ^, if 2 x 2 + 3 y 2 - 4 = o. 

air 

3. Find ^, if ax*f + ^ 2 = o. 

dx 

a -c" j dv -r x -f y . bx . h 

4. Find -^-, if — -^ H h - = o. 

rt^ x — y cy k 

5. Find -^-, if cos(.ri/) = .r. 

dx K ' J 

6. Find -^-, if log(^ 2 / 2 ) + .r 3 -f )p + 2 xy -\- a = o. 

dx 

7. Show § 112 geometrically. 



84 INFINITESIMAL CALCULUS 

114. Functions of many variables are peculiarly appli- 
cable in economic theory, though as yet they have been 
very little employed.* Many fallacies have been committed 
from lack of this more general conception of functional de- 
pendence, and from the tacit assumption that mere curves 
are capable of delineating any sort of quantitative relation. 
This is an error only one degree less flagrant than the errors 
of those whose sole mathematical idea is that of the con- 
stant quantity. 

* See, however, Edgeworth's Mathematical Psychics, 1881; the 
author's Mathematical Investigations in the Theory of Value and 
Prices, 1892; and Pareto's Cours d'econo77iie politique, 1896-7. 



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